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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 295, pp. 1--122.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/295\hfil Classification of quadratic differential systems]
{Geometric and algebraic classification of quadratic
differential systems with \\ invariant hyperbolas}

\author[R. D. S. Oliveira, A. C. Rezende, D. Schlomiuk, N. Vulpe 
\hfil EJDE-2017/295\hfilneg]
{Regilene D. S. Oliveira, Alex C. Rezende, \\
 Dana Schlomiuk, Nicolae Vulpe }

\address{Regilene D. S. Oliveira \newline
 Instituto de Ci\^encias Matem\'aticas e de Computa\c{c}\~ao,
 Universidade de S\~ao Paulo, Brazil}
\email{regilene@icmc.usp.br}

\address{Alex C. Rezende \newline
Universidade Federal de Santa Maria,
Campus Palmeira das Miss\~oes, Brazil}
\email{alexcrezende@gmail.com}

\address{Dana Schlomiuk \newline
 D\'epartement de Math\'ematiques et de Statistiques,
 Universit\'e de Montr\'eal, Canada}
\email{dasch@dms.umontreal.ca}

\address{Nicolae Vulpe \newline
Institute of Mathematics and Computer Science,
Academy of Sciences of Moldova}
\email{nvulpe@gmail.com}


\thanks{Submitted February 9, 2017. Published November 28, 2017.}
\subjclass[2010]{34C23, 34A34}
\keywords{Quadratic differential systems; algebraic solution;
\hfill\break\indent  configuration of algebraic solutions; affine invariant polynomials;
 group action}

\begin{abstract}
Let QSH be the whole class of non-degenerate planar
quadratic differential systems possessing at least one invariant
hyperbola. We classify this family of systems, modulo the
action of the group of real affine transformations and time
rescaling, according to their geometric properties encoded in the
configurations of invariant hyperbolas and invariant straight
lines which these systems possess. The classification is given
both in terms of algebraic geometric invariants and also in terms
of affine invariant polynomials. It yields a total of 205
distinct such configurations. We have 162 configurations for
the subclass QSH$_{(\eta>0)}$ of systems which possess three
distinct real singularities at infinity in $P_2(\mathbb{C})$, 
and 43 configurations for the subclass QSH$_{(\eta=0)}$ of systems
which possess either exactly two distinct real singularities at 
infinity or the line at infinity filled up with singularities. 
The algebraic classification, based on the invariant polynomials, 
is also an algorithm which makes it possible to verify for any given
real quadratic differential system if it has invariant hyperbolas or
not and to specify its configuration of invariant hyperbolas and
straight lines.
\end{abstract}

\maketitle
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{observation}[theorem]{Observation}
\allowdisplaybreaks
\renewcommand*{\figurename}{Diagram}

\tableofcontents

\section{Introduction and statement of the main results}

We consider planar polynomial differential systems which are systems of the form
\begin{equation}\label{sys:pol}
dx/dt=p(x,y),\quad dy/dt=q(x,y)
\end{equation}
where $p(x,y),q(x,y)$ are polynomial in $x,y$ with real
coefficients ($p,q\in \mathbb{R}[x,y]$) and their associated vector
fields
\begin{equation} \label{vectFild:X}
X=p(x,y)\frac{\partial}{\partial x} +q(x,y)\frac{\partial}{\partial y}.
\end{equation}
We call {\it degree} of such a system the number $\max(\deg(p),\deg(q))$.
In the case where the polynomials $p$ and $q$
are coprime we say that \eqref{sys:pol} is {\it non-degenerate}.

A real quadratic differential system is a polynomial differential system
of degree 2, i.e.
\begin{equation} \label{sys:QSgenCoef}
\begin{gathered}
 dx/dt=p_0+p_1(\tilde{a},x,y)+p_2(\tilde{a},x,y) \equiv p(\tilde{a},x,y),\\
 dy/dt=q_0+q_1(\tilde{a},x,y)+q_2(\tilde{a},x,y) \equiv q(\tilde{a},x,y)
\end{gathered}
\end{equation}
with $\max(\deg(p),\deg(q))=2$ and
\begin{gather*}
p_0=a,\quad p_1(\tilde{a},x,y)=cx+dy,\quad p_2(\tilde{a},x,y)=gx^2+2hxy+ky^2,\\
q_0=b,\quad q_1(\tilde{a},x,y)=ex+fy,\quad q_2(\tilde{a},x,y)=lx^2+2mxy+ny^2.
\end{gather*}
Here we denote by $\tilde{a}=(a,b,c,d,e,f,g,h,k,l,m,n)$ the
12-tuple of the coefficients of system \eqref{sys:QSgenCoef}. Thus
a quadratic system can be identified with a points $\tilde{a}$ in
$\mathbb{R}^{12}$.

We denote the class of all quadratic differential systems with QS.

Planar polynomial differential systems occur very often in various
branches of applied mathematics, in modeling natural phenomena,
for example, modeling the time evolution of interacting species in
biology and in chemical reactions and economics \cite{lotka,
volterra}, in astrophysics \cite{Chandrasekhar}, in the equations
of continuity describing the interactions of ions, electrons and
neutral species in plasma physics \cite{Roth}. Polynomial systems
appear also in shock waves, in neural networks, etc. Such
differential systems have also theoretical importance. Several
problems on polynomial differential systems, which were stated
more than one hundred years ago, are still open: the second part
of Hilbert's 16th problem stated by Hilbert in 1900 \cite{Hilbert:1902}, the problem
of algebraic integrability stated by Poincar\'e in 1891
\cite{Po2}, \cite{Po3}, the problem of the center stated by
Poincar\'e in 1885 \cite{Po1}, and problems on integrability
resulting from the work of Darboux \cite{Dar1878} published in
1878. With the exception of the problem of the center, which was
solved only for quadratic differential systems, all the other
problems mentioned above, are still unsolved even in the quadratic
case.

\begin{definition}[Darboux] \label{def1} \rm
An algebraic curve $f(x,y)=0$ where $f\in \mathbb{C}[x,y]$ is an invariant
curve of the planar polynomial system \eqref{sys:pol} if and only if there exists
a polynomial $k(x,y)\in \mathbb{C}[x,y]$ such that
$$
p(x,y)\frac{\partial f}{\partial x} + q(x,y)\frac{\partial
f}{\partial y}=f(x,y)k(x,y).
$$
\end{definition}

\begin{definition}[Darboux]  \label{def2} \rm
We call algebraic solution of a planar polynomial system an invariant
algebraic curve over $\mathbb{C}$ which is irreducible.
\end{definition}

One of our motivations in this article comes from integrability
problems related to the work of Darboux \cite{Dar1878}.

\begin{theorem}[Darboux] \label{thm1}
Suppose that a polynomial system \eqref{sys:pol} has $m$ invariant
algebraic curves $f_i(x,y)=0$, $i\leq m$, with $f_i\in \mathbb{C}[x,y]$
and with $m>n(n+1)/2$ where $n$ is the degree of the system. Then
there exist complex numbers $\lambda_1,\dots ,\lambda_m$ such that
$f_1^{\lambda_1}\dots f_m^{\lambda_m}$ is a first integral of the
system.
\end{theorem}

The condition in Darboux's theorem is only sufficient for Darboux integrability
(integrability in terms of invariant algebraic curves) and it is
not also necessary. Thus the lower bound on the number of
invariant curves sufficient for Darboux integrability stated in
the theorem of Darboux is larger than necessary. Darboux's theory
has been improved by including for example the multiplicity of the
curves (\cite{Llibre-Xian}). Also, the number of invariant algebraic curves needed was
reduced but by adding some conditions, in particular the condition
that any two of the curves be transversal. But a deeper
understanding about Darboux integrability is still lacking.
Algebraic integrability, which intervenes in the
open problem stated by Poincar\'e in 1891 (\cite{Po2} and \cite{Po3}),
and which means the existence of a
rational first integral for the system, is a particular case of
Darboux integrability.

\begin{theorem}[Jouanolou \cite{Jouanolou:1979}]\label{th:Jouanolou}
Suppose that a polynomial system \eqref{sys:pol}, defined by polynomials
$p(x,y)$, $q(x,y)\in \mathbb{C}[x,y]$, has $m$ invariant
algebraic curves $f_i(x,y)=0$, $i\leq m$, with $f_i\in \mathbb{C}[x,y]$
and with $m\geq n(n+1)/2+2$ where $n$ is the degree of the system. Then
the system has a rational first integral $h(x,y)/g(x,y)$ where
$h(x,y), g(x,y)\in \mathbb{C}[x,y]$.
\end{theorem}

To advance knowledge on algebraic or more generally Darboux
integrability it is necessary to have a large number of examples
to analyze. In the literature, scattered isolated examples were
analyzed but a more systematic approach was still needed.
Schlomiuk and Vulpe initiated a systematic program to construct
such a data base for quadratic differential systems. Since the
simplest case is of systems with invariant straight lines, their
first works involved only invariant lines for quadratic systems
(see \cite{SchVul04-QTDS,SchVul08-RMJM,SchVul08-NATMA,SchVul08-BASM,SchVul10-JFPTA}).
In this work we study a class of quadratic systems with invariant conics,
namely the class QSH
of non-degenerate (i.e. $p,q$ are relatively prime) quadratic
differential systems having an invariant hyperbola. Such systems
could also have some invariant lines and in many cases the
presence of these invariant curves turns them into Darboux
integrable systems. We always assume here that the systems
\eqref{sys:QSgenCoef} are non-degenerate because otherwise doing a
time rescaling, they can be reduced to linear or constant
systems. Under this assumption all the systems in QSH have
a finite number of finite singular points.

The irreducible affine conics over the field $\mathbb{R}$ are the hyperbolas,
ellipses and parabolas. One way to distinguish them is consider their points
at infinity (see \cite{Abhyankar:1988}). The term
hyperbola is used for a real irreducible affine conic which
has two real points at infinity. This distinguishes it from the
other two irreducible real conics: the parabola has just one real
point at infinity at which the multiplicity of intersection of the conic
with the line at infinity is two, and the ellipse which has two complex
points at infinity.

In the theory of Darboux the invariant algebraic
curves are considered (and rightly so) over the complex field $\mathbb{C}$. We
may extend the notion of hyperbola (parabola or ellipse) for conics
over $\mathbb{C}$. A hyperbola (respectively
parabola or ellipse) is an algebraic curve $C$ in $\mathbb{C}^2$, $C:f(x,y)=0$
with $f\in \mathbb{C}[x,y]$, $\deg(f)=2$ which is irreducible and which has two real
points at infinity (respectively
one real point at infinity with intersection multiplicity two, or two complex
(non-real) points at infinity).

\begin{observation} \label{obs1} \rm
 We draw attention to the fact
that if we have a curve $C:f(x,y)=0$ over $\mathbb{C}$ it could happen that multiplying
the equation by a number $\lambda\in \mathbb{C}^{*}=\mathbb{C}\backslash \{0\}$, the coefficients
of the new equation become real. In this case, to the equation $f(x,y)=0$
we can associate two curves: one real $\{(x,y)\in \mathbb{R}^2|\lambda f(x,y)=0\}$
and one complex $\{(x,y)\in \mathbb{C}^2|f(x,y)=0\}$. In particular if $f(x,y)\in \mathbb{R}[x,y]$
then we could talk about two curves, one in the real and the other in the complex
plane. If the coefficients of an algebraic curve $C:f(x,y)=0$ cannot be made real
by multiplication with a constant, then clearly to the equation $f(x,y)=0$
we can associate just one curve, namely the complex curve
$\{(x,y)\in \mathbb{C}^2|f(x,y)=0\}$.
\end{observation}

In this paper we consider real polynomial differential equations.
To each such a system of equations
 there corresponds the complex system with the same coefficients to which
we can apply the theory of Darboux using complex invariant algebraic curves.
 Some of these curves may turn out to be with real coefficients in which case
they also yield, as in the observation above, invariant algebraic curves in
$\mathbb{R}^2$ of the real differential system. It is one way, but not the only way,
in which the theory of Darboux yields applications to real systems.
 It is by juggling both with complex and real systems and their invariant
complex or real algebraic curves that we get a full understanding of the
classification problem we consider here.
In particular, apart from the hyperbolas (in the real plane) we shall encounter
 conics in the complex plane for which the coefficients cannot be made real
by the multiplication with a non-zero
complex constant and whose points at infinity are real and of course distinct,
just like for the (real) hyperbolas in the real plane. We call these conics
complex hyperbolas. These curves shed light on our classification problem.
Indeed, just as polynomials $g(x)\in \mathbb{R}[x]$ do not have always all their
roots in $\mathbb{R}$ but they factor into linear factors over $\mathbb{C}$ and full
understanding of the roots with their multiplicities only comes when we
consider them as elements of $\mathbb{C}[x]$, the complex invariant curves magnify
our understanding of the family QSH and help us in classifying QSH according
to the configurations of invariant hyperbolas and invariant lines.


Let us suppose that a polynomial differential system has an
algebraic solution $f(x,y)=0$ where $f(x,y)\in \mathbb{C}[x,y]$ is of
degree $n$,
$f(x,y)=a_0+a_{10}x+a_{01}y+\dots +a_{n0}x^n+a_{n-1,1}x^{n-1}y+\dots +a_{0n}y^n$
with $\hat a=(a_0,\dots ,a_{0n})\in \mathbb{C}^N$ where $N=(n+1)(n+2)/2$. We
note that the equation $\lambda f(x,y)=0$ where $\lambda\in
\mathbb{C}^{*}=\mathbb{C}\backslash \{0\}$ yields the same locus of
complex points in the plane as the locus induced by $f(x,y)=0$. So,
a curve of degree $n$ defined by $\hat a$ can be identified with a
point $[\hat a]=[a_0:a_{10}:\dots :a_{0n}]$ in $P_{N-1}(\mathbb{C})$. We say
that a sequence of degree $n$ curves $f_i(x,y)=0$ converges to a
curve $f(x,y)=0$ if and only if the sequence of points
$[a_i]=[a_{i0}:a_{i10}\dots :a_{i0n}]$ converges to $[\hat
a]=[a_0:a_{10}:\dots :a_{0n}]$ in the topology of $P_{N-1}(\mathbb{C})$.


On the class QS acts the group of real affine
transformations and time rescaling and because of this, modulo this
group action quadratic systems ultimately depend on five
parameters. In particular, restricting this group action on QSH, modulo this
action the QSH is a union of 1-dimensional, 2-dimensional and 3-dimensional
families of systems as it can be seen from the normal forms obtained
in \cite{Oliv-Rez-Vulpe} for this family.

We observe that if we rescale the time $t'=\lambda t$ by a
positive constant $\lambda$ the geometry of the systems
\eqref{sys:pol} does not change. So for our purposes we can
identify a system \eqref{sys:pol} of degree $n$ with a point in
$[a_0,a_{10},\dots ,b_{0n}]$ in $\mathbb{S}^{N-1}(\mathbb{R})$ with $N=(n+1)(n+2)$.

We compactify the space of all the polynomial differential systems
of degree $n$ on $\mathbb{S}^{N-1}$ with $N=(n+1)(n+2)$ by multiplying the
coefficients of each systems with $1/(\sum (a_{ij}^2+b_{ij}^2))^{1/2}$.

\begin{definition} \label{def:multiplicity} \rm
(1) We say that an invariant curve $\mathcal{L}:f(x,y)=0$, $f\in
\mathbb{C}[x,y]$ for a polynomial system (S) of degree $n$ has
\emph{multiplicity} $m$ if there exists a sequence of real
polynomial systems $(S_k)$ of degree $n$ converging to (S) in
the topology of $\mathbb{S}^{N-1}$, $N=(n+1)(n+2)$, such that each
$(S_k)$ has $m$ distinct invariant curves
$\mathcal{L}_{1,k}:f_{1,k}(x,y) = 0, \ldots ,
\mathcal{L}_{m,k}:f_{m,k}(x,y)=0$ over $\mathbb{C}$, $\deg(f)=\deg(f_{i,k})=r$,
converging to $\mathcal{L}$ as $k \to \infty$, in the topology of
$P_{R-1}(\mathbb{C})$, with $R=(r+1)(r+2)/2$ and this does not occur for
$m+1$.

(2) We say that the line at infinity $\mathcal{L}_{\infty}:Z=0$ of a
polynomial system (S) of degree $n$ has \emph{multiplicity}
$m$ if there exists a sequence of real polynomial systems $(S_k)$
of degree $n$ converging to (S) in the topology of
$\mathbb{S}^{N-1}$, $N=(n+1)(n+2)$, such that each $(S_k)$ has $m-1$
distinct invariant lines $\mathcal{L}_{1,k}:f_{1,k}(x,y) = 0,
\ldots ,\mathcal{L}_{m,k}:f_{m-1,k}(x,y)=0$ over $\mathbb{C}$, converging
to the line at infinity $\mathcal{L}_{\infty}$ as $k \to
\infty$, in the topology of $P_2(\mathbb{C})$ and this does not occur for
$m$.
\end{definition}

\begin{definition} \label{def4} \rm
(1) Suppose a planar polynomial system (S) has a finite number
of algebraic solutions $\mathcal{L}_i$, $i\leq k$, with
corresponding multiplicities $n_i$ and the line at infinity
 $\mathcal{L}_{\infty}$ is not filled up with singularities and it has
 multiplicity $n_{\infty}$. We call \emph{total
multiplicity} of these algebraic solutions, including the
multiplicity $n_{\infty}$ of the line at infinity $\mathcal{L}_{\infty}$,
the sum $TMC_{(S)}=n_1+\dots +n_k+n_{\infty}$.

(2) Suppose  system (S) has a finite number of real
distinct singularities $s_1,\dots ,s_l$, finite or infinite, which
are located on the algebraic solutions, and $s_1,\dots ,s_l$ have the
corresponding multiplicities $m_1,\dots ,m_l$. We call \emph{total
multiplicity of the real singularities on the invariant curves} of
(S) the sum $TMS_{(S)}=m_1+\dots +m_l$ and $TMS$ is the function
defined by this expression.
\end{definition}

An important ingredient in this work is the notion of
\emph{configuration of algebraic solution} of a polynomial
differential system. This notion appeared for the first time in
\cite{SchVul04-QTDS}.


\begin{definition} \label{def5} \rm
Consider a planar polynomial system which has a finite number of
algebraic solutions and a finite number of singular points,
finite or infinite. By \emph{configuration of algebraic
solutions} of this system we mean the set of algebraic solutions
over $\mathbb{C}$ of the system, each one of these curves endowed with its
own multiplicity and together with all the real singular points of
this system located on these curves, each one of these
singularities endowed with its own multiplicity.
\end{definition}


We may have two distinct systems which may be non-equivalent modulo the
action of the group but which may have ``the same configuration''
of invariant hyperbolas and straight lines. We need to say when two
configurations are ``the same'' or equivalent.

\begin{definition} \label{def6} \rm
Suppose we have two systems $(S_1),(S_2)$ in QSH with a finite number of
singularities, finite or infinite, a finite set of invariant hyperbolas
$\mathcal{H}_i:h_i(x,y)=0$, $i=1,\dots ,k$, of $(S_1)$
(respectively $\mathcal{H}'_i:h_i'(x,y)=0$, $i=1,\dots ,k$, of $(S_2)$)
and a finite set (which could also be empty) of invariant straight lines
$\mathcal{L}_j:f_j(x,y)=0$, $j=1,2,\dots k'$, of $(S_1)$
(respectively $\mathcal{L}_j':f_j' (x,y)=0$, $j=1,2,\dots k'$, of $(S_2)$).
We say that the two configurations $C_1,C_2$ of hyperbolas and lines of
these systems are equivalent if there is a one-to-one
correspondence $\phi_h$ between the hyperbolas
of $C_1$ and $C_2$ and a one-to-one correspondence $\phi_l$ between the
lines of $C_1$ and $C_2$ such that:

(i) the correspondences conserve the multiplicities of the hyperbolas
and lines (in case there are any) and also send a real invariant curve to a
real invariant curve and a complex invariant curve to a complex invariant curve;

(ii) for each hyperbola $\mathcal{H}: h(x,y)=0$ of $C_1$ (respectively each
line $\mathcal{L}: f(x,y)=0$) we have a one-to-one correspondence between the
real singular points on $\mathcal{H}$ (respectively on $\mathcal{L}$) and the real singular points on $\phi_h(\mathcal{H})$ (respectively
$\phi_l(\mathcal{L})$) conserving their multiplicities, their
location on branches of hyperbolas and their order on these branches
 (respectively on the lines);

(iii) Furthermore, consider the total curves
 $\mathcal{F}: \prod H_i(X,Y,X) \prod F_j(X,Y,Z)Z=0$ (respectively
$\mathcal{F}': \prod H'_i(X,Y,X) \prod F'_j(X,Y,Z)Z=0$ where
$H_i(X,Y,X)=0$, $F_j(X,Y,X)=0$ (respectively $H'_i(X,Y,X)=0$,
$F'_j(X,Y,X)=0$) are the projective completions of $\mathcal{H}_i$,
$\mathcal{L}_j$ (respectively $\mathcal{H}'_i$, $\mathcal{L}'_j$).
Then, there is a correspondence $\psi$ between the singularities
of the curves $\mathcal{F}$ and $\mathcal{F}'$ conserving their
multiplicities as singular points of the total curves.
\end{definition}

In the family QSH we also have cases where we have an infinite number of hyperbolas.
Thus, according to the theorem of Jouanolou (Theorem \ref{th:Jouanolou}),
we have a rational first integral. In this case the multiplicity of a hyperbola
in the family is either considered to be undefined or we may say that this
multiplicity is infinite. Such situations occur either when we have $(i)$ a
finite number of singularities, finite or infinite, or $(ii)$ an infinite
number of singularities which could only be at infinity (recall that the
systems in QSH are non-degenerate). In both cases however we show that we have
a finite number of affine invariant straight lines with finite multiplicities.
In fact it was proved in \cite{SchVul08-JDDE} that all quadratic systems which
have the line at infinity filled up with singularities have affine invariant
straight lines of total multiplicity three.
Furthermore, the multiplicities of singularities of the systems are finite
in the case $(i)$ and this is also true in case $(ii)$ if we only take into
consideration the affine lines. We therefore can talk about the
\emph{configuration of affine invariant lines associated to the system}.
Two such configurations of affine invariant lines $C_{1L}$, $C_{2L}$ associated to
systems $(S_1),(S_2)$ are said to be equivalent if and only if there is a
one-to-one
correspondence $\phi_l$ between the
lines of $C_{1L}$ and $C_{2L}$ such that:
\begin{enumerate}
\item[(i)] the correspondence conserves the multiplicities of lines and also
sends a real invariant line to a real invariant line and a complex invariant
line to a complex invariant line;
\item[(ii)] for each line $\mathcal{L}: f(x,y)=0$ we have a one-to-one
correspondence between the real singular points on $\mathcal{L}$ and
the real singular points on $\phi_l(\mathcal{L})$) conserving their
 multiplicities and their order on the lines.
\end{enumerate}
We use this to extend our previous definition further above to cover these cases.

\begin{definition} \label{def7} \rm
Suppose we have two systems $(S_1),(S_2)$ in QSH with a finite number of finite
singularities and an infinite number of invariant hyperbolas of $(S_1)$
(respectively an infinite number of hyperbolas of $(S_2)$).
Suppose we have a non-empty finite set of affine invariant straight
lines $\mathcal{L}_j:f_j(x,y)=0$, $j=1,2,\dots k$, of $(S_1)$
(respectively $\mathcal{L}_j':f_j' (x,y)=0$, $j=1,2,\dots k$, of $(S_2)$).
We now consider only the two configurations $C_{1L}$, $C_{2L}$ of invariant
affine lines of $(S_1)$, $(S_2)$ associated to the systems.
We say that the two configurations $C_{1L}$, $C_{2L}$ are
\emph{equivalent with respect to the hyperbolas of the systems}
if and only if $(i)$ they are equivalent as configurations of invariant
lines and in addition the following property (ii)
 is satisfied: we take any hyperbola $\mathcal{H}: h(x,y)=0$ of $(S_1)$
and any hyperbola $\mathcal{H}': h'(x,y)=0$ of $(S_2)$. Then, we must
have a one-to-one correspondence between the real singular points of the
system $(S_1)$ located on $\mathcal{H}$ and of real singular points of the system
$(S_2)$ located on $\mathcal{H}'$, conserving their multiplicities and their
location and order on branches. Furthermore, consider the curves
 $\mathcal{F}: \prod h(x,y)\prod f_j(x,y)=0$ and
$\mathcal{F}': \prod h'(x,y)\prod f'_j(x,y)=0$.
Then we have a one-to-one correspondence between the singularities of the
 curve $\mathcal{F}$ with those of the curve $\mathcal{F}'$ conserving their
multiplicities as singular points of these curves.
\end{definition}

It can be easily shown that the definition above is independent of the choice
 of the two hyperbolas $\mathcal{H}: h(x,y)=0$ of $(S_1)$ and
$\mathcal{H}': h'(x,y)=0$ of $(S_2)$.

In \cite{Oliv-Rez-Vulpe} the authors provide necessary and
sufficient conditions for a non-degenerate quadratic differential
system to have at least one invariant hyperbola and these
conditions are expressed in terms of the coefficients of the
systems. In this paper we denote by QSH$_{(\eta>0)}$ the family of
non-degenerate quadratic systems in QSH possessing three distinct real
singularities at infinity and by QSH$_{(\eta=0)}$
the systems in QSH possessing either exactly two
distinct real singularities at infinity or the line at infinity
filled up with singularities. We classify
these families of systems, modulo the action of the group of real
affine transformations and time rescaling, according to their
geometric properties encoded in the configurations of invariant
hyperbolas and/or invariant straight lines which these systems
possess.


As we want this classification to be intrinsic, independent of the normal
form given to the systems, we use here geometric invariants and invariant polynomials
for the classification. For example, it is clear that the configuration of
algebraic solutions of a system in QSH is an affine invariant. The classification
is done according to the configurations of invariant hyperbolas and straight
lines encountered in systems belonging to QSH. We put in the same equivalence class systems which have equivalent configurations of invariant hyperbolas and/or lines. In particular the notion of
multiplicity in Definition \ref{def:multiplicity} is invariant
under the group action, i.e. if a quadratic system $S$ has an
invariant curve $\mathcal{L}=0$ of multiplicity $m$, then each
system $S\,'$ in the orbit of $S$ under the group action has a
corresponding invariant line $\mathcal{L}'=0$ of the same
multiplicity $m$. To distinguish configurations of algebraic solutions
we need some geometric invariants, and we also use invariant polynomials
both of which are
introduced in our Section \ref{sec:auxil results}.

\begin{theorem} \label{mainthm}
Consider the class {\rm QSH} of all
non-degenerate quadratic differential systems \eqref{sys:QSgenCoef} possessing
an invariant hyperbola.

{\rm (A)} This family is classified according
 to the configurations of invariant hyperbolas and of invariant straight lines
 of the systems, yielding 205 distinct such configurations, 162 of which belong
 to the class {\rm QSH}$_{(\eta>0)}$ and 43 to {\rm QSH}$_{(\eta=0)}$.
 This geometric classification is described in Theorems \ref{th:Main1} and
 \ref{th:Main2}.

{\rm (B)} Using invariant polynomials, we obtain the bifurcation diagram
in the space $\mathbb{R}^{12}$ of the coefficients of the system in {\rm  QS} according
to their configurations of invariant hyperbolas and invariant straight
lines (this diagram is presented in part (B) of Theorems \ref{th:Main1}
and \ref{th:Main2}). Moreover, this diagram gives an algorithm to compute
the configuration of a system with an invariant hyperbola for any
quadratic differential system, presented in any normal form.
\end{theorem}

The article is organized as follows:
In Section \ref{sec:auxil results} we define all the geometric and algebraic
invariants used in the paper and we introduce the basic auxiliary results
we need for the proof of our theorems. In Section 3 we consider the class
{\rm QSH}$_{(\eta>0)}$ of all non-degenerate quadratic differential systems
\eqref{sys:QSgenCoef} possessing three distinct real singularities at infinity
and we classify this family according to the geometric configurations of
invariant hyperbolas and invariant straight lines which they possess.
We also give their bifurcation diagram in the 12-dimensional space $\mathbb{R}^{12}$
of their coefficients, in terms of invariant polynomials.
In section 4 we consider the class QSH$_{(\eta=0)}$ of all
non-degenerate quadratic differential systems
\eqref{sys:QSgenCoef} possessing an invariant hyperbola and either exactly
two distinct real singularities at infinity or the line at infinity filled up with
singularities. We classify this family according to the geometric
configurations of invariant hyperbolas and invariant straight lines
which they possess. We also give their bifurcation diagram in the
12-dimensional space $\mathbb{R}^{12}$ of their coefficients,
in terms of invariant polynomials.
In section \ref{sec:conclusion}  we give some concluding comments,
stressing the fact that the bifurcation diagrams in $\mathbb{R}^{12}$ give
us an algorithm to compute the
configuration of a system with an invariant hyperbola for any
system presented in any normal form.

\section{Basic concepts and auxiliary results}\label{sec:auxil results}

In this section we define all the invariants we use in the Main Theorem
and we state some auxiliary results. A quadratic system possessing an invariant
hyperbola could also possess invariant lines. We classified the
systems possessing an invariant hyperbola in terms of their
configurations of invariant hyperbolas and invariant lines. Each
one of these invariant curves has a multiplicity in the sense of
Definition \ref{def:multiplicity} (see also
in~\cite{ChrLliPer2007}). We encode this picture in the
multiplicity divisor of invariant hyperbolas and lines. We first
recall the algebraic-geometric definition of an $r$-cycle on an
irreducible algebraic variety of dimension $n$.

\begin{definition} \label{def8} \rm
Let $V$ be an irreducible algebraic variety of dimension $n$ over
a field $K$. A cycle of dimension $r$ or $r$-cycle on $V$ is a
formal sum $\Sigma_{W}n_{W} W$, where $W$ is a subvariety of $V$ of
dimension $r$ which is not contained in the singular locus of $V$,
$n_W\in \mathbb{Z}$, and only a finite number of $n_W$'s are non-zero. We
call \emph{degree of an $r$-cycle} the sum $\Sigma n_W$. An $(n-1)$-cycle
is called a \emph{divisor}.
\end{definition}

\begin{definition} \label{def9} \rm
Let $V$ be an irreducible algebraic variety over a field $K$. The
\emph{support of a cycle $C$} on $V$ is the set
$\operatorname{supp}(C) =\{W|n_W\neq 0\}$. We denote by $Max(C)$ the maximum
value of the coefficients $n_W$ in $C$. For every $m\leq Max(C)$
 let $s(m)$ be the number of the coefficients $n_W$ in $C$ which are equal to $m$.
We call \emph{type of the cycle $C$} the set of ordered couples
$(s(m),m)$
where $1\leq m\leq Max(C)$.
\end{definition}

Clearly the degree and the type of an $r$-cycle are invariant under
the action of the group of real affine transformations and time rescaling.

For a non-degenerate polynomial differential systems (S) possessing
a finite number of algebraic solutions $f_i(x,y)=0$, $f_i(x,y)\in \mathbb{C}$,
each with multiplicity
$n_i$ and a finite number of singularities at infinity, we define the
\emph{algebraic solutions divisor} on
the projective plane, $ICD=\Sigma_{n_i}n_i\mathcal
C_i+n_{\infty}\mathcal{L}_{\infty}$ (also called the \emph{invariant curves divisor})
where $\mathcal C_i: F_i(X,Y,Z)=0$ are the projective completions
of $f_i(x,y)=0$, $n_i$ is the multiplicity of the curve $\mathcal
C_i=0$ and $n_{\infty}$ is the multiplicity of the line at
infinity $\mathcal{L}_{\infty}:Z=0$. It is well known (see
\cite{Art_Llibre}) that the maximum number of invariant straight
lines, including the line at infinity, for polynomial systems of
degree $n\ge2$ is $3n$.

In the case we consider here, we have a particular instance of the
divisor (CD because the invariant curves will be invariant
hyperbolas and invariant lines of a quadratic differential system,
in case these are in finite number. In case we have an infinite
number of hyperbolas we use only the invariant lines to construct
the divisor.

Another ingredient of the configuration of algebraic solutions are
the real singularities situated on these curves. We also need to use
here the notion of multiplicity divisor of real singularities of a
system, located on the algebraic solutions of the system.

\begin{definition}\label{def:ICD,MSoc} \rm

(1) Suppose a real quadratic system has a finite number of
 invariant hyperbolas $\mathcal{H}_i:h_i(x,y)=0$ $i=1,\dots ,k$ and a finite
 number of affine invariant lines $\mathcal{L}_j:f_j(x,y)=0$, $j=1,\dots ,l$. We denote
 the line at infinity $\mathcal{L}_{\infty}:Z=0$. Let us assume
 that on the line at infinity we have a finite number of
 singularities. The divisor of invariant hyperbolas
 and invariant lines on the complex projective plane of the system
 is the following:
 \[
 ICD =n_1 \mathcal{H}_1+\ldots + n_k\mathcal{H}_k+m_1\mathcal{L}_1+\ldots + m_k\mathcal{L}_l+
 m_\infty\mathcal{L}_\infty,
 \]
where $n_i$ (respectively
$m_j$) is the multiplicity of the hyperbola $\mathcal{H}_i$
(respectively $m_j$ of the line $\mathcal{L}_j$), and $m_{\infty}$ is the multiplicity of $\mathcal{L}_{\infty}$. We
also mark the complex (non-real) invariant hyperbolas (respectively lines)
denoting them by $\mathcal{H}_i^C$ (respectively
$\mathcal{L}_i^C$). We define the total multiplicity TMH of
invariant hyperbolas as the sum $\sum_i{n_i}$ and the total
multiplicity TML of invariant line as the sum $\sum_i{m_i}$.
We denote by IHD (respectively ILD) the invariant hyperbolas
divisor (respectively the invariant lines divisor) i.e.
$IHD=n_1 \mathcal{H}_1+\ldots + n_k\mathcal{H}_k$ (respectively
$ILD=m_{\infty}\mathcal{L}_{\infty}+m_1\mathcal{L}_1+\ldots + m_l\mathcal{L}_l$).

(2) The zero-cycle on the real projective plane, of real singularities of
 a system \eqref{sys:QSgenCoef} located on the configuration of invariant lines
 and invariant hyperbolas, is given by:
 \[
 M S_{0C} = r_1U_1+\ldots + r_l {U}_l+v_1s_1+\ldots + v_n {s}_n,
 \]
where $U_i$ (respectively $s_j$) are all the real infinite (respectively
finite) such singularities of the system and $r_i$ (respectively $v_j$)
are their corresponding multiplicities.
\end{definition}

In the family QSH we have configurations which have an infinite
number of hyperbolas. These are of two kinds: those with a finite
number of singular points at infinity, and those with
the line at infinity filled up with singularities. To distinguish
these two cases we define $|\operatorname{Sing}_{\infty}|$ to be the cardinality
of the set of singular points at infinity of the systems. In the first case we
have $|\operatorname{Sing}_{\infty}|=2$ or 3, and in the second case
 $|\operatorname{Sing}_{\infty}|$
is the continuum and we simply write $|\operatorname{Sing}_{\infty}|=\infty$.
Since in both cases the systems admit a finite number of affine
invariant straight lines we can use them to distinguish the
configurations.

\begin{definition} \label{def11} \rm
(1)
 In case we have an infinite number of hyperbolas and just two or three singular
points at infinity but we have a finite number of invariant straight lines
we define $ ILD = m_1\mathcal{L}_1+\ldots +
m_k\mathcal{L}_k+m_\infty\mathcal{L}_\infty$ (see Definition
\ref{def:ICD,MSoc});

(2) In case we have an infinite number of hyperbolas, the line at infinity is filled
up with singularities and we have a finite number of affine lines, we define
$ILD =m_1\mathcal{L}_1+\ldots + m_k\mathcal{L}_k$.
\end{definition}

Attached to the divisors and the zero-cycle we defined,
we have their \emph{types} which are clearly affine invariants.
So although the cycles ICD and MS$_{0C}$ are not themselves
affine invariants, they are used in the classification because we can read on them
several specific invariants, such as for example their types, TMS, TMC, etc.

The above defined divisor (CD and zero-cycle MS$_{0C}$ contain
several invariants such as the number of invariant lines and their
total multiplicity TML, the number of invariant hyperbolas (in
case these are in finite number) and their total multiplicity
TMH, the number of complex invariant hyperbolas of a real
system, etc.

There are two compactifications
which intervene in the classification of QSH according to
the configurations of the systems: the compactification in
the Poincar\'e disk and the compactification of its associated
foliation with singularities on the real projective plane
$P_2(\mathbb{R})$. We also have the compactification of its associated
 (complex) foliation with singularities on the complex projective plane.
Each one of these compactifications plays a role in the classification.
In the compactified system the line at
infinity of the affine plane is an invariant line. The system may
have singular points located at infinity which are not points of
intersection of invariant curves, points also denoted by $U_r$.

The real singular points at infinity (respectively the real finite singular
 points) which are intersection point of two or more
invariant algebraic curves are denoted by $\overset{j}{U}_r$
(respectively by $\overset{j}{s}_r$), where
$j\in\{h,l,hh,hl,ll,llh^{\infty}, \ldots \}$. Here $h$ (respectively
$l,hh,hl,ll,llh^{\infty}, \ldots$) means that the intersection of the
infinite line with a hyperbola (respectively with a line, or with
two hyperbolas, or with a hyperbola and a line, or with two lines,
or with two lines and an infinite number of hyperbolas etc.).
In other words, whenever the symbol $h^{\infty}$ appears in the divisor
MS$_{0C}$ it means that the singularity lies on an infinite number of hyperbolas.

Suppose we have a finite number of real invariant hyperbolas and real
invariant straight lines of a system (S) and that they
are given by equations $f_i(x,y)=0$, $i\in \{1,2,\dots ,k\}$,
$f_i\in \mathbb{R}[x,y]$. Let us denote by $F_i(X,Y,Z)=0$ the projection
completion of the invariant curves $f_i=0$ in $P_2(\mathbb{R})$.

\begin{definition} \label{def12} \rm
The total invariant curve of the system (S) in QSH, on $P_2(\mathbb{C})$, is the curve
$\mathcal{T}(S): \prod{F_i(X,Y,Z)}Z=0$.
\end{definition}

We use the above notion to define the \emph{basic curvilinear
polygons determined by the total curve $\mathcal{T}(S)$}. Consider
the Poincar\'e disk and remove from it the (real) points of the
total curve $\mathcal{T}(S)$. We are left with a certain number of
2-dimensional connected components.

\begin{definition} \label{def13} \rm
We call basic polygon determined by $\mathcal{T}(S)$ the
closure in the Poincar\'e disk of anyone of these components associated
to $\mathcal{T}(S)$.
\end{definition}

Although a basic polygon is a 2-dimensional object, we shall think
of it as being just its border.

The singular points of the system (S) situated on $T(S)$
 are of two kinds: those which are simple (or smooth) points
of $\mathcal{T}(S)$ and those which are multiple points of
$\mathcal{T}(S)$.

\begin{remark}\label{rem:ss-points} \rm
To each singular point of the system we have its associated
multiplicity as a singular point of the system. In addition, we
also have the multiplicity of these points as points on the total
curve $\mathcal{T}(S)$. Through a singular point of
the systems there may pass several of the curves $F_i=0$ and
$Z=0$. Also we may have the case when this point is a singular
point of one or even of several of the curves in case we work with
invariant curves with singularities. This leads to the
multiplicity of the point as point of the curve $\mathcal{T}(S)$.
The simple points of the curve $\mathcal{T}(S)$ are those of multiplicity one.
They are also the smooth points of this curve.
\end{remark}

The real singular points of the system which are simple points of
$\mathcal{T}(S)$ are useful for defining some geometrical
invariants, helpful in the geometrical classification, besides
those which can be read from the zero-cycle defined further above.

We now introduce the notion of \emph{minimal proximity polygon}
of a singular point of the total curve. This notion plays a major
role in the geometrical classification of the systems.

\begin{definition} \label{def14} \rm
Suppose a system (S) has a finite number of singular points and a
 finite number of invariant hyperbolas and straight lines.
Let $p$ be a real singular point of a system lying on $\mathcal{T}(S)$
and in the Poincar\'e disk. Then $p$ may belong to several basic
polygons. We call \emph{minimal proximity polygon of $p$} a basic
polygon on which $p$ is located and which has the minimum number
of vertices, among the basic polygons to which $p$ belongs. In
case we have more than one polygon with the minimum number of
vertices, we take all such polygons as being \emph{minimal
proximity polygons of $p$}.
\end{definition}

\begin{remark} \label{rmk2} \rm
We observe that for systems in QSH we have a
basic polygon located in the finite plane only in one case
(Config. H.36) and the polygon is a triangle. All other polygons
 have at least one vertex at infinity.
\end{remark}


For a configuration $C$, consider for each real singularity $p$ of
the system which is a simple point of the curve $\mathcal{T}(S)$,
its minimal proximity basic polygons. We construct some formal
finite sums attached to the Poincar\'e disk, analogs of the
algebraic-geometric notion of divisor on the projective plane. For
this we proceed as follows:

We first list all real singularities of the systems on the
Poincar\'e disk which are simple points ($ss$-points) of the total curve. In
case we have such points $U_i$'s located on the line at infinity,
we start with those points which are at infinity. We obtain a list
$U_1,\ldots,U_n,s_1,\ldots,s_k$, where $s_i$'s are finite points.
Associate to $U_1,\ldots,U_n$ their minimal proximity polygons
$\mathcal{P}_1,\ldots,\mathcal{P}_m$. In case some of them
coincide we only list once the polygons which are repeated. These
minimal proximity polygons may contain some finite points from the
list $s_1,\ldots,s_k$. We remove all such points from this list.
Suppose we are left with the finite points $s_{j_1},\ldots,s_{j_r}$. For
these points we associate their corresponding minimal proximity
polygons. We observe that for a point $s_{j_i}$ we may have two
minimal proximity polygons in which case we consider only the
minimal proximity polygon which has the maximum number of
singularities $s_j$, simple points of the total curve ($ss$ -points). If the two
polygons have the same maximum number of simple $ss$-points then we take the two
of them. We obtain a list of polygons and we retain from this list
only that polygon (or those polygons) which have the maximal
number of $ss$-points and add these polygons to the list
$\mathcal{P}_1,\ldots,\mathcal{P}_m$. We remove all the
$ss$-points which appear in this list of polygons from the list of
points $s_{j_1},\ldots,s_{j_r}$ and continue the same process until
there are no points left from the sequence $s_1,\ldots,s_k$ which
have not being included or eliminated. We thus end up with a list
$\mathcal{P}_1,\ldots,\mathcal{P}_l$ of proximity polygons which we denote
by $\mathcal{P}(C)$.

\begin{definition} \label{def:PD} \rm
We denote by PD the proximity ``divisor'' of the Poincar\'e disk
\[
{\rm PD}=v_1\mathcal{P}_1+\cdots+v_r\mathcal{P}_l,
\]
associated to the list $\mathcal{P}(C)$ of the
minimal proximity polygons of a configuration, where
$\mathcal{P}_i$ are the minimal proximity polygons from this list
and $v_i$ are their corresponding number of vertices.
\end{definition}

We used the word \emph{divisor} of the Poincar\'e disk in analogy
with divisor on the projective plane, also thinking of polygons as the borders of
the 2-dimensional polygons.

The next divisor considers the
proximity polygons in PD but only the ones attached to the
finite singular points of the system which are simple points on
the total curve. So this time we start with all such points
$s_1,\ldots,s_k$ and build up the divisor like we did before. The
result is called ``\emph{the proximity divisor of the real finite
singular points of the systems, simple points of the total curve}''
and we denote it by PD$_f$.

We also define a divisor on the Poincar\'e disk which encodes the
way the minimal proximity polygons intersect the line at
infinity.

\begin{definition} \label{def16} \rm
We denote by PD$_\infty$ the ``divisor'' of the Poincar\'e disk
encoding the way the proximity polygons occurring in PD intersect the
infinity and define it as
\[
PD_\infty=\sum_\mathcal{P} n_\mathcal{P}\mathcal{P},
\]
where $\mathcal{P}$ is a proximity polygon occurring in PD and
$n_\mathcal{P}$ is $3$ if $\mathcal{P}$ has one of its sides on
the line at infinity, it is $2$ if $\mathcal{P}$ has only two
vertices on the line at infinity, it is $1$ if only one of its
vertices lies on the line at infinity and it is $0$ if $\mathcal{P}$ is finite.
\end{definition}

\begin{definition} \label{def17} \rm
For a proximity polygon $\mathcal{P}$ we introduce the multiplicity divisor
\[
m \mathcal{P}=\sum m(v) \ v,
\]
where $v$ is a vertex of $\mathcal{P}$ and $m(v)$ is the
multiplicity of the singular point $v$ of the system.
\end{definition}


In case a configuration $C$ has an invariant hyperbola
$\mathcal{H}$ and an invariant line $\mathcal{L}$, we define the
following invariant I which helps us decide the type of their
intersection.

\begin{definition} \label{def18} \rm
Suppose we have an invariant line $\mathcal{L}$ and an invariant
hyperbola $\mathcal{H}$ of a polynomial differential system
(S). We define the invariant $I$ attached to the couple
$\mathcal{L},\mathcal{H}$ as being: $0$ if and only if
$\mathcal{L}$ intersects $\mathcal{H}$ in two complex non-real
points; $1$ if and only if $\mathcal{L}$ is tangent to $\mathcal{H}$;
$21$ if and only if $\mathcal{L}$ intersects $\mathcal{H}$
in two real points and both these points lie on only one branch of
the hyperbola; $22$ if and only if $\mathcal{L}$ intersects
$\mathcal{H}$ in two real points and these points lie on distinct
branches of the hyperbola. In case for a configuration $C$ we
have several hyperbolas ${\mathcal{H}_i}$, $i\in \{1,2,\dots ,r\}$ and
an invariant line $\mathcal{L}$, then $I=\{I(\mathcal{L},
\mathcal{H}_1),I(\mathcal{L},\mathcal{H}_2),\dots ,I(\mathcal{L},\mathcal{H}_r)\}$.
\end{definition}


\begin{definition} \label{def19} \rm
We define a function $O$ (for ``order''), $O: {\rm QSH}\rightarrow \{1,0,-1\}$
as follows: Suppose a system (S) in
QSH has two singular points at infinity, one simple
$U_1$ and the other double $U_2$. Suppose the system has only one
invariant hyperbola and only two real finite singular points $s_1$
and $s_2$ lying on a branch of the invariant hyperbola connecting
$U_1$ with $U_2$ such that $s_2$ is double and $s_1$ is simple.
We have only two possibilities: either the segment of hyperbola
connecting the two double singularities $U_2$ and $s_2$ contains
$s_1$ in which case we write $O(S)=1$ or it does not contain $s_1$
and then we write $O(S)=0$. In case we have a configuration where
this specific situation does not occur we write $O(S) =-1$.
\end{definition}


A few more definitions and results which play an important role in
the proof of the part (B) of the Main Theorem are needed. We do not prove these
results here but we indicate where they can be found.

Consider the differential operator $\mathcal{L}= x\cdot L_2
-y\cdot L_1$ constructed in \cite{Blt_Vlp_DEDS} and acting on
$\mathbb R[\tilde a,x,y]$, where
\begin{align*}
 L_1=& 2a_{00}\frac{\partial}{\partial a_{10}} +
 a_{10}\frac{\partial}{\partial a_{20}} +
 \frac{1}{2}a_{01}\frac{\partial}{\partial a_{11}}
+2b_{00}\frac{\partial}{\partial b_{10}} +
 b_{10}\frac{\partial}{\partial b_{20}} +
 \frac{1}{2}b_{01}\frac{\partial}{\partial b_{11}},\\
 L_2= &2a_{00}\frac{\partial}{\partial a_{01}} +
 a_{01}\frac{\partial}{\partial a_{02}} +
 \frac{1}{2}a_{10}\frac{\partial}{\partial a_{11}}
+2b_{00}\frac{\partial}{\partial b_{01}} +
 b_{01}\frac{\partial}{\partial b_{02}} +
 \frac{1}{2}b_{10}\frac{\partial}{\partial b_{11}}.
\end{align*}
Using this operator and the affine invariant
$\mu_0=\operatorname{Res}_x\big(p_2(\tilde a,x,y),q_2(\tilde a,x,y)\big)/y^4 $
we construct the following polynomials
\[
\mu_i(\tilde a,x,y) =\frac{1}{i!} \mathcal{L}^{(i)}(\mu_0), \
i=1,\dots,4,\] where
$\mathcal{L}^{(i)}(\mu_0)=\mathcal{L}(\mathcal{L}^{(i-1)}(\mu_0))$ and
$\mathcal{L}^{(0)}(\mu_0)=\mu_0$.

These polynomials are in fact comitants of systems
\eqref{sys:QSgenCoef} with respect to the group $GL(2,\mathbb R)$ (see
\cite{Blt_Vlp_DEDS}). Their geometrical meaning is revealed in the
next lemma.

\begin{lemma}[\cite{Baltag,Blt_Vlp_DEDS}] \label{lem:mu_i-ISPs}
Assume that a quadratic system {\rm (S)} with coefficients $\tilde a$
belongs to the family \eqref{sys:QSgenCoef}. Then:

(i) Let $\lambda$ be an integer such that $\lambda\leq 4$. The total
multiplicity of all finite singularities of this system equals $4-\lambda$
if and only if for every $i\in\{0,1,\ldots,\lambda-1\}$ we have
$\mu_i(\tilde{a},x,y)=0$ in the ring $\mathbb{R}[x,y]$ and
$\mu_\lambda(\tilde{a},x,y)\ne0$. In this case, the factorization
$\mu_{\lambda}(\tilde{a},x,y)={\prod_{i=1}^{\lambda}(u_ix-v_iy)\ne0}$
over $\mathbb{C}$ indicates the coordinates $[v_i:u_i:0]$ of those
finite singularities of the system (S) which ``have gone'' to infinity.
Moreover, the number of distinct factors in this factorization is less
than or equal to three (the maximum number of infinite singularities
of a quadratic system in the projective plane) and the multiplicity of
each one of the factors $u_ix-v_iy$ gives us the number of the finite
singularities of the system (S) which have coalesced with the
infinite singular point $[v_i:u_i:0]$.

(ii) System (S) is degenerate (i.e. $\gcd(P,Q)\ne{\rm const}$) if and only
if $\mu_i(\tilde{a},x,y)=0$ in $\mathbb{R}[x,y]$ for every $i=0,1,2,3,4$.
\end{lemma}

The following zero-cycle on the complex plane was introduced in
\cite{Llibre-Schlomiuk:2004} based on previous work in \cite{Sch-Pal:2001}.

\begin{definition} \label{def20} \rm
We define $\mathcal D_{\mathbb{C}^2}(S)=\sum_{s\in \mathbb{C}^2}n_ss$
where $n_s$ is the intersection multiplicity at $s$ of the curves
$p(x,y)=0$, $q(x,y)=0$, $p,q$ being the polynomials defining the equations
\eqref{sys:pol} for system (S).
\end{definition}

\begin{proposition}[\cite{Vlp-NATMA}] \label{prop:number-FSPs}
The form of the zero-cycle $\mathcal D_{\mathbb{C}^2}(S)$ for non-degenerate
quadratic systems \eqref{sys:QSgenCoef} is determined by the
corresponding conditions indicated in Table 1, where we
write $p+q+r^c+s^c$ if two of the finite points, i.e. $r^c, s^c$,
are complex but not real, and
\begin{equation} \label{Pol:D,T,R,e.al.}
\begin{aligned}
{\rm D}&=\Big[3\big((\mu_3,\mu_3)^{(2)},\,\mu_2\big)^{(2)}
- \big(6\mu_0\mu_4-3\mu_1\mu_3+\mu_2^2,\ \mu_4)^{(4)}\Big]/48,\\
{\rm P}&=12\mu_0\mu_4-3\mu_1\mu_3+\mu_2^2,\\
{\rm R}&=3\mu_1^2-8\mu_0\mu_2,\\
{\rm S}&={\rm R}^2-16\mu_0^2{\rm P},\\
{\rm T}&=18\mu_0^2(3\mu_3^2-8\mu_2\mu_4)+2\mu_0(2\mu_2^3-9\mu_1\mu_2\mu_3+27\mu_1^2\mu_4)-
{\rm P R},\\
{\rm U}&=\mu_3^2-4\mu_2\mu_4,\\
{\rm V}&= \mu_4.
\end{aligned}
\end{equation}
\end{proposition}

\begin{table}[ht]
\caption{Number and multiplicity of the finite
singular points of  QS}
\begin{center}
\scriptsize \renewcommand{\arraystretch}{1.2}
\begin{tabular}{|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|}

\hline
 \,No. & {\begin{tabular}{c} Zero--cycle\\ $\mathcal D_{\mathbb{C}^2}(S)$\end{tabular}}
 &   \begin{tabular}{c}Invariant \\ criteria \end{tabular}
 & \,No.
 & {\begin{tabular}{c} Zero--cycle\\ $\mathcal D_{\mathbb{C}^2}(S)$\end{tabular}}
 &   \begin{tabular}{c}Invariant \\ criteria \end{tabular}  \\
\hline
 1 & $p+q+r+s$ &$\begin{array}{c} \mu_0\ne0, {\rm D}<0,\\[-.2mm] {\rm R}>0, {\rm S}>0\end{array}$&
 10 & \,$p+q+r $\, &$\begin{array}{c} \mu_0=0, {\rm D}<0, {\rm R}\ne0\end{array}$\\
\hline
 2 & \,$p+q+r^c+s^c$\, &$\begin{array}{c} \mu_0\ne0, {\rm D}>0\end{array}$&
 11 & $p+q^c+r^c $ &$\begin{array}{c} \mu_0=0, {\rm D}>0, {\rm R}\ne0\end{array}$\\
\hline
 \raisebox{-0.7em}[0pt][0pt]{3}
 & \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}p^{\,c}+q^c+r^c+s^c\end{array}$}
 &$\begin{array}{c} \mu_0\ne0, {\rm D}<0, {\rm R}\le0\end{array}$ &
 \raisebox{-0.7em}[0pt][0pt]{12} & \raisebox{-0.7em}[0pt][0pt]{$2p+q$} &
 \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c} \mu_0={\rm D}=0, {\rm P}{\rm R}\ne0\end{array}$}\\
 \cline{3-3}
 & & $\begin{array}{c} \mu_0\ne0, {\rm D}<0, {\rm S}\le0\end{array}$ & & &\\
\hline
 {4} & {$2p+q+r $} & {$\begin{array}{c} \mu_0\ne0, {\rm D}=0, {\rm T}<0 \end{array}$}&
 13 & $3p $ &$\begin{array}{c} \mu_0 = {\rm D} = {\rm P} = 0, {\rm R}\ne0\end{array}$\\
\hline
 5 & $2p+q^c+r^c$ &$\begin{array}{c} \mu_0\ne0, {\rm D}=0, {\rm T}>0\end{array}$&
 14 & $p+q$ &$\begin{array}{c} \mu_0={\rm R}=0, {\rm P}\ne0,\\[-0.5mm]{\rm U}>0\end{array}$\\
\hline
 6 & \,$2p+2q $\, &$\begin{array}{c} \mu_0\ne0, {\rm D}={\rm T}=0,\\[-0.5mm] {\rm P}{\rm R}>0\end{array}$ &
 15 & \,$p^{\,c}+q^c $\, &$\begin{array}{c} \mu_0={\rm R}=0, {\rm P}\ne0,\\[-0.5mm]{\rm U}<0\end{array}$\\
\hline
 7 & $2p^{\,c}+2q^c$ &$\begin{array}{c} \mu_0\ne0, {\rm D}={\rm T}=0,\\[-0.5mm] {\rm P}{\rm R}<0\end{array}$&
 16 & $2p $ &$\begin{array}{c} \mu_0={\rm R}=0, {\rm P}\ne0,\\[-0.5mm]{\rm U}=0\end{array}$\\
\hline
 8 & \,$3p+q $\, &$\begin{array}{c} \mu_0\ne0, {\rm D}={\rm T}=0,\\[-0.5mm]{\rm P}=0,{\rm R}\ne0\end{array}$ &
 17 & \,$p $\, &$\begin{array}{c} \mu_0={\rm R}={\rm P}=0,\\[-0.5mm] {\rm U}\ne0 \end{array}$\\
\hline
 9 & $4p $ &$\begin{array}{c} \mu_0\ne0, {\rm D}={\rm T}=0,\\[-0.5mm] {\rm P}={\rm R}=0\end{array}$&
 18 & $0$ &$\begin{array}{c} \mu_0={\rm R}={\rm P}=0,\\[-0.5mm]{\rm U}=0, {\rm V}\ne0\end{array}$\\
\hline
\end{tabular}
\end{center}
 \end{table}


The next result is stated in \cite{Oliv-Rez-Vulpe} and it gives us
the necessary and sufficient conditions for the existence of at
least one invariant hyperbola for non-degenerate systems
\eqref{sys:QSgenCoef} and also their multiplicities. The invariant polynomials
which appears in the statement of the next theorem and in the
corresponding diagrams are constructed in \cite{Oliv-Rez-Vulpe}
and we present them further below.

\begin{theorem}[\cite{Oliv-Rez-Vulpe}] \label{trm:exist-hyp}
{\rm (A)} The conditions $\gamma_1=\gamma_2=0$ and
either $\eta\ge0$, $M\ne0$ or $C_2=0$ are necessary for a
quadratic system in the class {\rm QS} to possess at least one
invariant hyperbola.

{\rm (B)} Assume that for a system in the class {\rm QS} the condition
$\gamma_1=\gamma_2=0$ is satisfied.
\begin{itemize}

 \item[(B1)] If $\eta>0$ then the necessary and sufficient
conditions for this system to possess at least one invariant
hyperbola are given in Diagram \ref{diagr:eta-poz},
where we can also find the number and multiplicity of such
hyperbolas.

\item[(B2)] In the case $\eta=0$ and either $M\ne0$ or $C_2=0$
the corresponding necessary and sufficient conditions for this
system to possess at least one invariant hyperbola are given
in Diagram \ref{diagr:eta=0}, where we can also find the
number and multiplicity of such hyperbolas.
\end{itemize}

{\rm (C)} The Diagrams \ref{diagr:eta-poz} and
\ref{diagr:eta=0} actually contain the global bifurcation diagram
in the 12-dimensional space of parameters of the coefficients of the systems
belonging to family {\rm QS}, which possess at least one
 invariant hyperbola. The corresponding conditions
are given in terms of invariant polynomials with respect to the
group of real affine transformations and time rescaling.
\end{theorem}


\begin{remark} \label{rem:Hp} \rm
An invariant hyperbola is denoted by $\mathcal{H}$ if it is real
and by $\overset{c}{\mathcal{H}}$ if it is complex. In
the case we have two such hyperbolas then it is necessary to
distinguish whether they have parallel or non-parallel asymptotes
in which case we denote them by $\mathcal{H}^p$
($\overset{c}{\mathcal{H}^p}$) if their asymptotes are
parallel and by $\mathcal{H}$ if there exists at least one pair of
non-parallel asymptotes. We denote by $\mathcal{H}_k$ ($k=2,3$) a
hyperbola with multiplicity $k$; by $\mathcal{H}_2^p$ a double
hyperbola, which after perturbation splits into two
$\mathcal{H}^p$; and by $\mathcal{H}_3^p$ a triple hyperbola
which splits into two $\mathcal{H}^p$ and one $\mathcal{H}$.
\end{remark}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.9\textwidth]{fig1} % Diagram-Hyp-eta-poz-New.eps
\end{center}
\caption{Existence of invariant hyperbolas: the case
$\eta>0$} \label{diagr:eta-poz}
\end{figure}

\begin{figure}[htb]
\begin{center}
 \includegraphics[width=0.9\textwidth]{fig2} % Diagram-Hyp-eta-zero-New.eps
\end{center}
\caption{Existence of invariant hyperbolas: the case
$\eta=0$}\label{diagr:eta=0}
\end{figure}

Following \cite{Oliv-Rez-Vulpe} we present here the invariant
polynomials which according to Diagrams
\ref{diagr:eta-poz} and \ref{diagr:eta=0} are responsible for the
existence and the number of invariant hyperbolas which systems
\eqref{sys:QSgenCoef} could possess.

First we single out the following five polynomials, basic
ingredients in constructing invariant polynomials for systems
\eqref{sys:QSgenCoef}:
\begin{equation} \label{expr:Ci,Di}
\begin{gathered}
 C_i(\tilde a,x,y)=yp_i(x,y)-xq_i(x,y),\ (i=0,1,2)\\
 D_i(\tilde a,x,y)=\frac{\partial p_i}{\partial x}+
 \frac{\partial q_i}{\partial y},\ (i=1,2).
\end{gathered}
\end{equation}
As it was shown in \cite{Sib1} these polynomials of degree one in
the coefficients of systems \eqref{sys:QSgenCoef} are
$GL$-comitants of these systems.
 Let $f,g\in\mathbb{R}[\tilde a,x,y]$ and
\[ % \label{def:transv}
 (f,g)^{(k)}=
 \sum_{h=0}^k (-1)^h \binom{k}{h}
 \frac{\partial^k f}{ \partial x^{k-h}\partial y^h}\
 \frac{\partial^k g}{ \partial x^h\partial y^{k-h}}.
\]
The polynomial $(f,g)^{(k)}\in \mathbb{R}[\tilde a,x,y] $ is called
\emph{the transvectant of index $k$ of $(f,g)$} (cf.
\cite{GraYou41,Olver}).

\begin{theorem}[see  \cite{Vul86-Book}] \label{th:Vlp}
Any $GL$-comitant of systems \eqref{sys:QSgenCoef} can be
constructed from the elements \eqref{expr:Ci,Di} by using the
operations: $+,\, -,\,\times$, and by applying the differential
operation $(*,* )^{(k)}$.
\end{theorem}

\begin{remark} \label{rmk4} \rm
We point out that the elements \eqref{expr:Ci,Di}
generate the whole set of $GL$-comitants and hence also the set
of affine comitants as well as the set of $T$-comitants and
$CT$-comitants (see \cite{SchVul04-QTDS} for detailed
definitions).
\end{remark}

We construct the following $GL$-comitants of the second degree
with respect to the coefficients of the initial systems
\begin{equation} \label{expr:Ti}
\begin{gathered}
 T_1=\left(C_0,C_1\right)^{(1)},\quad
 T_2=\left(C_0,C_2\right)^{(1)},\quad
 T_3=\left(C_0,D_2\right)^{(1)},\\
 T_4=\left(C_1,C_1\right)^{(2)},\quad
 T_5=\left(C_1,C_2\right)^{(1)},\quad
 T_6=\left(C_1,C_2\right)^{(2)},\\
 T_7=\left(C_1,D_2\right)^{(1)},\quad
 T_8=\left(C_2,C_2\right)^{(2)},\quad
 T_9=\left(C_2,D_2\right)^{(1)}.
\end{gathered}
\end{equation}



Using these $GL$-comitants as well as the polynomials
\eqref{expr:Ci,Di} we construct additional invariant
polynomials. To be able to directly calculate the values of the
invariant polynomials we need, for every canonical system
we define here a family of $T$-comitants expressed
through $C_i$ $(i=0,1,2)$ and $D_j$ $(j=1,2)$:
\begin{gather*}
\hat A= \left(C_1,T_8-2T_9+D_2^2\right)^{(2)}/144,\\
\begin{aligned}
\widehat D&= \Big[2C_0(T_8-8T_9-2D_2^2)+C_1(6T_7-T_6
 -\left(C_1,T_5\right)^{(1)}\\
&\quad +6D_1(C_1D_2 -T_5)-9D_1^2C_2\Big]/36,
\end{aligned}\\
\widehat E= \left[D_1(2T_9-T_8)-3\left(C_1,T_9\right)^{(1)}
 -D_2(3T_7+D_1D_2)\right]/72,\\
\begin{aligned}
\widehat F&=
\Big[6D_1^2(D_2^2-4T_9)+4D_1D_2(T_6+6T_7)
 +48C_0\left(D_2,T_9\right)^{(1)} -9D_2^2T_4 \\
&\quad + 288 D_1\widehat E
 -24\big(C_2,\widehat D\big)^{(2)} + 120\big(D_2,\widehat
 D\big)^{(1)} \\
&\quad - 36C_1\left(D_2,T_7\right)^{(1)}
 + 8D_1\left(D_2,T_5\right)^{(1)}\Big]/144,
\end{aligned} \\
\begin{aligned}
\widehat B
&=\Big\{16D_1\left(D_2,T_8\right)^{(1)}\left(3C_1D_1-2C_0D_2+4T_2\right)
 +32C_0\left(D_2, T_9\right)^{(1)}\big(3D_1D_2 \\
&\quad -5T_6+9T_7\big) +2\left(D_2, T_9\right)^{(1)}\big(27C_1T_4-18C_1D_1^2\\
&\quad -32D_1T_2+32\left(C_0, T_5\right)^{(1)}\big)
  +6\left(D_2, T_7\right)^{(1)}\big[8C_0(T_8-12T_9)\\
&\quad -12C_1(D_1D_2+T_7)+ D_1(26C_2D_1+32T_5)
 + C_2(9T_4+96T_3)\big]\\
&\quad +6\left(D_2,  T_6\right)^{(1)}[32C_0T_9-C_1(12T_7+52D_1D_2)
 -32C_2D_1^2] \\
&\quad +48D_2\left(D_2, T_1\right)^{(1)}\left(2D_2^2-T_8\right)
 -32D_1T_8\left(D_2, T_2\right)^{(1)}+9D_2^2T_4\left(T_6-2T_7\right)\\
&\quad -16D_1\left(C_2, T_8\right)^{(1)}\left(D_1^2+4T_3 \right)
 +12D_1\left(C_1, T_8\right)^{(2)}\left(C_1D_2-2C_2D_1 \right) \\
&\quad +6D_1D_2T_4\left(T_8-7D_2^2-42T_9\right)
 +12D_1\left(C_1,T_8\right)^{(1)}\left(T_7+2D_1D_2\right)\\
&\quad +96D_2^2\left[D_1\left(C_1, T_6\right)^{(1)}
 +D_2\left(C_0, T_6\right)^{(1)}\right]
 -16D_1D_2T_3\left(2D_2^2+3T_8\right) \\
&\quad -4D_1^3D_2\left(D_2^2+3T_8+6T_9\right)
 +6D_1^2D_2^2\left(7T_6+2T_7\right)
 -252D_1D_2T_4T_9\big\} /(2^{8}3^{3}),
\end{aligned}\\
\widehat K= (T_8+4T_9+4D_2^2)/72, \quad
\widehat H= (8T_9-T_8+2D_2^2)/72,\quad
\widehat N=4\widehat K-4\widehat H.
\end{gather*}
These polynomials in addition to \eqref{expr:Ci,Di} and
\eqref{expr:Ti} will serve as bricks in constructing affine
invariant polynomials for systems \eqref{sys:QSgenCoef}.

Using the above bricks, the following 42 affine invariants $A_1,\ldots,A_{42}$
are constructed from the minimal polynomial basis of affine invariants
 up to degree 12. This fact was proved in \cite{Bul_Tim}.
\begin{gather*}
 A_1 = \hat A,\quad
 A_2 = (C_2, \widehat D)^{(3)}/12, \\
 A_3 = \big[ C_2, D_2)^{(1)}, D_2\big)^{(1)}, D_2\big)^{(1)}/48, \quad
 A_4 = (\widehat H, \widehat H)^{(2)}, \\
 A_5 = (\widehat H, \widehat K)^{(2)}/2,\quad
 A_6 = (\widehat E, \widehat H)^{(2)}/2,\\
 A_7 = \big[ C_2, \widehat E)^{(2)}, D_2\big)^{(1)}/8, \quad
 A_8 = \big[\widehat D, \widehat H)^{(2)}, D_2\big)^{(1)}/8, \\
 A_9  = \big[\widehat D, D_2)^{(1)}, D_2\big)^{(1)}, D_2\big)^{(1)}/48, \quad
 A_{10} = \big[\widehat D, \widehat K)^{(2)}, D_2\big)^{(1)}/8,\\
 A_{11} = (\widehat F, \widehat K)^{(2)}/4,\quad
 A_{12} = (\widehat F, \widehat H)^{(2)}/4,\\
 A_{13} = \big[C_2, \widehat H)^{(1)}, \widehat H\big)^{(2)}, D_2\big)^{(1)}/24,\quad
 A_{14} = (\widehat B, C_2)^{(3)}/36,\\
 A_{15} = (\widehat E, \widehat F)^{(2)}/4, \quad
 A_{16} = \big[\widehat E, D_2)^{(1)}, C_2\big)^{(1)},\widehat K\big)^{(2)}/16, \\
 A_{17} = \big[\widehat D,\widehat D)^{(2)},D_2\big)^{(1)},D_2\big)^{(1)}/64, \quad
 A_{18} = \big[\widehat D,\widehat F)^{(2)}, D_2\big)^{(1)}/16,\\
 A_{19} = \big[\widehat D,\widehat D)^{(2)},\widehat H\big)^{(2)}/16, \quad
 A_{20} = \big[C_2,\widehat D)^{(2)}, \widehat F\big)^{(2)}/16,\\
 A_{21} = \big[\widehat D, \widehat D)^{(2)}, \widehat K\big)^{(2)}/16, \quad
 A_{22} = \frac{1}{1152}\big[C_2, \widehat D)^{(1)}, D_2\big)^{(1)},
   D_2\big)^{(1)}, D_2\big)^{(1)} D_2\big)^{(1)},\\
 A_{23} = \big[\widehat F, \widehat H)^{(1)}, \widehat K\big)^{(2)}/8,\quad
 A_{24} = \big[C_2, \widehat D)^{(2)}, \widehat K\big)^{(1)},
 \widehat H\big)^{(2)}/32,\\
 A_{25} = \big[\widehat D, \widehat D)^{(2)}, \widehat E\big)^{(2)}/16,\quad
 A_{26} =\big(\widehat B, \widehat D \big))^{(3)}/36,\\
 A_{27} = \big[\widehat B, D_2)^{(1)}, \widehat H\big)^{(2)}/24,\quad
 A_{28} = \big[C_2,\widehat K)^{(2)},\widehat D\big)^{(1)},\widehat E\big)^{(2)}/16,\\
 A_{29} = \big[\widehat D, \widehat F)^{(1)}, \widehat D\big)^{(3)}/96,\quad
 A_{30} = \big[C_2,\widehat D)^{(2)},\widehat D\big)^{(1)},\widehat D\big)^{(3)}/288,\\
 A_{31} = \big[\widehat D,\widehat D)^{(2)},\widehat K\big)^{(1)},
 \widehat H\big)^{(2)}/64,\quad
 A_{32} = \big[\widehat D, \widehat D)^{(2)}, D_2\big)^{(1)},
 \widehat H\big)^{(1)}, D_2\big)^{(1)}/64,\\
 A_{33} = \big[\widehat D, D_2)^{(1)}, \widehat F\big)^{(1)},
  D_2\big)^{(1)}, D_2\big)^{(1)}/128,\\
 A_{34} = \big[\widehat D, \widehat D)^{(2)}, D_2\big)^{(1)},
 \widehat K\big)^{(1)}, D_2\big)^{(1)}/64,\\
 A_{35} = \big[\widehat D, \widehat D)^{(2)}, \widehat E\big)^{(1)},
 D_2\big)^{(1)}, D_2\big)^{(1)}/128,\quad
 A_{36} = \big[\widehat D,\widehat E)^{(2)},\widehat D\big)^{(1)},
 \widehat H\big)^{(2)}/16,\\
 A_{37} = \big[\widehat D,\widehat D)^{(2)},\widehat D\big)^{(1)},
 \widehat D\big)^{(3)}/576,\quad
 A_{38} = \big[C_2,\widehat D)^{(2)}, \widehat D\big)^{(2)},
 \widehat D\big)^{(1)}, \widehat H\big)^{(2)}/64,\\
 A_{39} = \big[\widehat D,\widehat D)^{(2)},\widehat F\big)^{(1)},
 \widehat H\big)^{(2)}/64,\quad
 A_{40} = \big[\widehat D,\widehat D)^{(2)},\widehat F\big)^{(1)},
\widehat K\big)^{(2)}/64,\\
 A_{41} = \big[C_2,\widehat D)^{(2)}, \widehat D\big)^{(2)},
\widehat F\big)^{(1)}, D_2\big)^{(1)}/64,\quad
 A_{42} = \big[\widehat D,\widehat F)^{(2)},\widehat F\big)^{(1)},
D_2\big)^{(1)}/16.
\end{gather*}
In the above list, the bracket ``$[$'' is used in order to avoid
placing the otherwise necessary up to five parentheses ``$($''.

Using the elements of the minimal polynomial basis given above the
following affine invariant polynomials were constructed in
\cite{Oliv-Rez-Dana-Vulpe}.
\begin{equation}
\gamma_1(\tilde a)= A_1^2 (3A_6 + 2 A_7) - 2 A_6( A_8 +A_{12}), \label{pageref:gamma1}
\end{equation}
\begin{align*}
\gamma_2(\tilde a)
&=9 A_1^2 A_2 (23252 A_3 + 23689 A_4) -
 1440 A_2 A_5 (3 A_{10} + 13 A_{11}) \\
&\quad - 1280 A_{13} (2 A_{17} + A_{18} + 23 A_{19}
 - 4 A_{20}) - 320 A_{24} (50 A_8 + 3 A_{10} \\
&\quad+ 45 A_{11} - 18 A_{12}) +
 120 A_1 A_6 (6718 A_8 + 4033 A_9 + 3542 A_{11}
 + 2786 A_{12}) \\
&\quad + 30 A_1 A_{15}(14980 A_3 - 2029 A_4 - 48266 A_5)
 - 30 A_1 A_7 (76626 A_1^2 \\
&\quad - 15173 A_8 + 11797 A_{10} + 16427 A_{11} - 30153 A_{12})\\
&\quad + 8 A_2 A_7 (75515 A_6  - 32954 A_7)
 + 2 A_2 A_3 (33057 A_8 - 98759 A_{12}) \\
&\quad - 60480 A_1^2 A_{24} + A_2 A_4 (68605 A_8
 - 131816 A_9 + 131073 A_{10} + 129953 A_{11}) \\
&\quad - 2 A_2 (141267 A_6^2 - 208741 A_5 A_{12} + 3200 A_2 A_{13}),
\end{align*}
\begin{align*}
\gamma_3(\tilde a)
&= 843696 A_5 A_6 A_{10} +  A_1 (-27 (689078 A_8 + 419172 A_9 - 2907149 A_{10} \\
&\quad - 2621619 A_{11}) A_{13} -  26 (21057 A_3 A_{23} + 49005 A_4 A_{23} \\
&\quad  - 166774 A_3 A_{24} +  115641 A_4 A_{24})),
\end{align*}
\begin{align*}
\gamma_4(\tilde a)
&=-9 A_4^2 (14 A_{17} + A_{21}) +
 A_5^2 (-560 A_{17} - 518 A_{18} + 881 A_{19} - 28 A_{20} \\
&\quad + 509 A_{21}) - A_4 (171 A_8^2 + 3 A_8 (367 A_9 - 107 A_{10}) +
 4 (99 A_9^2 + 93 A_9 A_{11} \\
&\quad + A_5 (-63 A_{18}  - 69 A_{19} + 7 A_{20} + 24 A_{21}))) + 72 A_{23} A_{24},
\end{align*}
\begin{align*}
\gamma_5(\tilde a)
&=-488 A_2^3 A_4 + A_2 (12 (4468 A_8^2 + 32 A_9^2
 - 915 A_{10}^2 + 320 A_9 A_{11} -  3898 A_{10} A_{11} \\
&\quad - 3331 A_{11}^2 +  2 A_8 (78 A_9 + 199 A_{10} + 2433 A_{11})) +
 2 A_5 (25488 A_{18} \\
&\quad - 60259 A_{19} - 16824 A_{21}) + 779 A_4 A_{21}) + 4 (7380 A_{10} A_{31} \\
&\quad - 24 (A_{10} + 41 A_{11}) A_{33} +
 A_8 (33453 A_{31} + 19588 A_{32} - 468 A_{33} - 19120 A_{34}) \\
&\quad + 96 A_9 (-A_{33} + A_{34}) + 556 A_4 A_{41} -
 A_5 (27773 A_{38} + 41538 A_{39}\\
&\quad  - 2304 A_{41} + 5544 A_{42})),
\end{align*}
\[
\gamma_6(\tilde a)=2 A_{20} - 33 A_{21},
\]
\begin{align*}
\gamma_7(\tilde a)
&= A_1 (64 A_3 - 541 A_4) A_7 + 86 A_8 A_{13} + 128 A_9 A_{13} - 54 A_{10} A_{13}
 - 128 A_3 A_{22} \\
&\quad + 256 A_5 A_{22} + 101 A_3 A_{24} - 27 A_4 A_{24},
\end{align*}
\begin{align*}
\gamma_8(\tilde a)
&=3063 A_4 A_9^2 - 42 A_7^2 (304 A_8 + 43 (A_9 - 11 A_{10}))
 - 6 A_3 A_9 (159 A_8 + 28 A_9\\
&\quad  + 409 A_{10}) + 2100 A_2 A_9 A_{13} + 3150 A_2 A_7 A_{16}
 + 24 A_3^2 (34 A_{19}\\
&\quad  - 11 A_{20}) + 840 A_5^2 A_{21} - 932 A_2 A_3 A_{22}
 + 525 A_2 A_4 A_{22} \\
&\quad + 844 A_{22}^2 - 630 A_{13} A_{33},
\end{align*}
\begin{gather*}
\gamma_9(\tilde a)=2 A_8 - 6 A_9 + A_{10},\quad
\gamma_{10}(\tilde a)=3 A_8 + A_{11},\\
\gamma_{11}(\tilde a)=-5 A_7 A_8 + A_7 A_9 + 10 A_3 A_{14},\quad
\gamma_{12}(\tilde a)=25 A_2^2 A_3 + 18 A_{12}^2,\\
\gamma_{13}(\tilde a)=A_2,\quad
\gamma_{14}(\tilde a)=A_2 A_4 + 18 A_2 A_5 - 236 A_{23} + 188 A_{24},
\end{gather*}
\begin{align*}
\gamma_{15}(\tilde a,x,y)
&=144 T_1 T_7^2 - T_1^3 (T_{12} + 2
T_{13}) - 4 (T_9 T_{11} + 4 T_7 T_{15} + 50 T_3 T_{23} + 2 T_4 T_{23} \\
&\quad + 2 T_3 T_{24} + 4 T_4 T_{24}),
\end{align*}
\begin{gather*}
\gamma_{16}(\tilde a,x,y)=T_{15},\quad
\gamma_{17}(\tilde a,x,y)=T_{11}+12T_{13},\\
\tilde\gamma_{18}(\tilde a,x,y)=C_1 (C_2,C_2)^{(2)} - 2C_2(C_1,C_2)^{(2)},\\
\tilde\gamma_{19}(\tilde a,x,y)=D_1 (C_1,C_2)^{(2)} - ((C_2,C_2)^{(2)},C_0)^{(1)},\\
\delta_1(\tilde a)=9 A_8 + 31 A_9 + 6 A_{10},\quad
\delta_2(\tilde a)=41 A_8 + 44 A_9 + 32 A_{10},\\
\delta_3(\tilde a)=3 A_{19} - 4 A_{17},\quad
\delta_4(\tilde a)=-5 A_2 A_3 + 3 A_2 A_4 + A_{22},\\
\delta_5(\tilde a)=62 A_8 + 102 A_9 - 125 A_{10},\quad
\delta_6(\tilde a)=2 T_3 + 3 T_4,\\
\beta_1(\tilde a)=3 A_1^2 - 2 A_8 - 2 A_{12},\quad
\beta_2(\tilde a)=2 A_7 - 9 A_6,\\
\beta_3(\tilde a)=A_6,\quad
\beta_4(\tilde a)=-5 A_4 + 8 A_5,\\
\beta_5(\tilde a)=A_4,\quad
\beta_6(\tilde a)=A_1,\\
\beta_7(\tilde a)=8 A_3 - 3 A_4 - 4 A_5,\quad
\beta_8(\tilde a)=24 A_3 + 11 A_4 + 20 A_5,\\
\beta_9(\tilde a)=-8 A_3 + 11 A_4 + 4 A_5,\quad
\beta_{10}(\tilde a)=8 A_3 + 27 A_4 - 54 A_5,\\
\beta_{11}(\tilde a,x,y)=T_1^2 - 20 T_3 - 8 T_4,\quad
\beta_{12}(\tilde a,x,y)=T_1,\\
\beta_{13}(\tilde a,x,y)=T_3,
\end{gather*}

\begin{gather*}
\begin{aligned}
\mathcal{R}_1(\tilde a)
&=-2 A_7 (12 A_1^2 + A_8 + A_{12}) + 5 A_6 (A_{10} + A_{11}) - 2 A_1 (A_{23}\\
&\quad - A_{24}) + 2 A_5 (A_{14} + A_{15}) + A_6 (9 A_8 + 7 A_{12}),
\end{aligned}\\
\mathcal{R}_2(\tilde a)=A_8 + A_9 - 2 A_{10},\quad
\mathcal{R}_3(\tilde a)=A_9,\\
\mathcal{R}_4(\tilde a)=-3 A_1^2 A_{11} + 4 A_4 A_{19},\\
\begin{aligned}
\mathcal{R}_5(\tilde a,x,y)
&=(2 C_0 (T_8 - 8 T_9 - 2 D_2^2) +  C_1 (6 T_7 - T_6) - (C_1,T_5)^{(1)} \\
&\quad + 6 D_1 (C_1 D_2 - T_5) - 9 D_1^2 C_2),
\end{aligned}\\
\begin{aligned}
\mathcal{R}_6(\tilde a)
&=-213 A_2 A_6 + A_1 (2057 A_8 - 1264 A_9 + 677 A_{10}
 + 1107 A_{12}) \\
&\quad + 746 (A_{27} - A_{28}),
\end{aligned}\\
\mathcal{R}_7(\tilde a)=-6 A_7^2 - A_4 A_8 + 2 A_3 A_9 - 5 A_4 A_9 + 4 A_4 A_{10}
 - 2 A_2 A_{13},\\
\mathcal{R}_8(\tilde a)=A_{10},\quad
\mathcal{R}_9(\tilde a)=-5 A_8 + 3 A_9,\\
\mathcal{R}_{10}(\tilde a)=7 A_8 + 5 A_{10} + 11 A_{11},\quad
\mathcal{R}_{11}(\tilde a,x,y)=T_{16},
\end{gather*}

\begin{gather*}
\chi_A^{(1)}(\tilde a)= A_6(A_1 A_2 - 2 A_{15}) (3 A_1^2 - 2 A_8 - 2 A_{12}),\\
\begin{aligned}
\chi_B^{(1)}(\tilde a)
&= A_7 \big[41 A_1 A_2 A_3 + 846 A_6 A_9 -252 A_6 A_{10} + 3798 A_6 A_{11}
 - 2 A_7 (6588 A_1^2 \\
&\quad - 830 A_8 + 265 A_{10} + 366 A_{11} - 156 A_{12})  + 1098 A_6 A_{12} \\
&\quad + 983 A_3 A_{14} - 1548 A_4 A_{14} - 365 A_3 A_{15}
 + 1350 A_4 A_{15} + 1550 A_2 A_{16} \\
&\quad - 1350 A_1 A_{23}\big],
\end{aligned}\\
\begin{aligned}
\chi_C^{(1)}(\tilde a)
&= \theta \beta_1 \beta_3 \big[8 A_1 ( 42 A_{23} - 24 A_2 A_3 + 59 A_2 A_5) +
 A_6 ( 2196 A_1^2 + 384 A_9 + 24 A_{10} \\
&\quad + 360 A_{11} - 432 A_{12}) + 4 A_7 (123 A_8  -  61 A_{10}
  -  23 A_{11} + 123 A_{12}) \\
&\quad + 8 (2 A_4 A_{14}  -  34 A_5 A_{15}  -  19 A_2 A_{16})\big],
\end{aligned}\\
\widetilde \chi_D^{(1)}(\tilde a)= -378 A_1^2 + 213 A_8 + 40 A_9
 - 187 A_{10} - 205 A_{11} + 317 A_{12},\\
\begin{aligned}
 \chi_E^{(1)}(\tilde a)
&= 48 A_6 (65 A_9 - 54 A_{10} - 27 A_{11}) -
 16 A_7 (774 A_1^2 - 382 A_8 + 263 A_{10} \\
&\quad + 129 A_{11}  - 360 A_{12}) +  72 A_4 (23 A14 + 3 A15) - 16 A_3 (163 A14 \\
&\quad + 185 A15)  - 1792 A_2 A_{16} + 16 A_1 (54 A_2 A_5 - 173 A_{22} + 27 A_{24}),
\end{aligned}\\
 \chi_F^{(1)}(\tilde a)= \theta \beta_1 \beta_3 \big[A_7 (2 A_5 - A_4) - 2 A_3 A_6
 \big],\quad
 \chi_G^{(1)}(\tilde a)= 12 A_3 - 7 A_4,\\
\chi_A^{(2)}(\tilde a)= A_4 ( 5 A_8 - 18 A_1^2 - A_{10} - 3 A_{11} + 9 A{12}),\\
 \chi_B^{(2)}(\tilde a)= A_3 (2 A_8 - 6 A_1^2 - A_9 + A_{10} - A_{11} + 3 A_{12}),
\\
 \begin{aligned}
\chi_A^{(3)}(\tilde a)
&= 49071656765835 A_1^6 +  27 A_1^4 (1344257279043 A_{11}
 - 1270094588593 A_{12})\\
&\quad + 3 A_1^2 (176071859457 A_2^2 A_4 + 2042424190056 A_{11}^2
 -\! 4553853105234 A_{11} A_{12}\\
&\quad + 2056276619466 A_{12}^2 +
 221071597034 A_5 A_{18} - 539155411551 A_5 A_{19} \\
&\quad + 65833344676 A_5 A_{20} + 26464141896 A_4 A_{21} +
 303070135713 A_5 A_{21}\\
&\quad - 137515925820 A_2 A_{23}) +
 1048 (35846142 A_2^2 A_4 A_{11} - 163576560 A_{11}^3 \\
&\quad - 21276288 A_2^2 A_4 A_{12} - 195478380 A_{11}^2 A_{12} +
 325223640 A_{11} A_{12}^2 \\
&\quad - 93862680 A_{12}^3 + 782460 A_4 A_8 A_{20} +  26186136 A_2 A_8 A_{22}\\
&\quad + 42548200 A_2 A_9 A_{22} -  2682720 A_5^2 A_{29}
 - 83946780 A_2 A_5 A_{31} \\
&\quad +  429178020 A_2 A_5 A_{32}  - 204768603 A_2 A_4 A_{34}
 -  125823390 A_2 A_5 A_{34}),
\end{aligned}\\
\begin{aligned}
\chi_B^{(3)}(\tilde a)
&= 10687627614087 A_1^6 -  36 A_1^2 A_{11} (57734730901 A_{11}\\
&\quad - 18520980346 A_{12})  - 54 A_1^4 (29889576561 A_{11} \\
&\quad + 85579885241 A_{12}) - 1848441298229 A_4 A_8 A_{19} \\
&\quad  - 995417129104 A_4 A_{10} A_{19} + 139152650610 A_5 A_{10} A_{19} \\
&\quad - 854619791782 A_4 A_{11} A_{19} - 234092667978 A_5 A_{11} A_{19} \\
&\quad - 1064773031314 A_4 A_{12} A_{19}   - 1538921088774 A_5 A_{12} A_{19}\\
&\quad - 200109956062 A_4 A_8 A_{20} - 33399158264 A_4 A_{10} A_{20} \\
&\quad + 1182168636 A_5 A_{10} A_{20}-  33699561192 A_4 A_{11} A_{20} \\
&\quad + 359794764 A_5 A_{11} A_{20} - 150658987068 A_4 A_{12} A_{20}\\
&\quad - 97478758260 A_5 A_{12} A_{20} -  1043930677997 A_4 A_8 A_{21} \\
&\quad - 381285679090 A_4 A_{10} A_{21} - 266080146306 A_5 A_{10} A_{21} \\
&\quad - 340140897016 A_4 A_{11} A_{21} - 373227206190 A_5 A_{11} A_{21}\\
&\quad - 763104633190 A_4 A_{12} A_{21} - 470713035534 A_5 A_{12} A_{21},
\end{aligned}\\
\begin{aligned}
\chi_C^{(3)}(\tilde a)
&= -(30838311945 A_1^2 A_2^2 A_4 + 2760800121876 A_1^2 A_8^2\\
&\quad + 7697984307234 A_1^2 A_8 A_9 + 3201113344320 A_1^2 A_9^2 \\
&\quad - 1697507613684 A_1^2 A_8 A_{10} + 31825111584 A_2^2 A_4 A_{11}\\
&\quad - 695990880 A_1^2 A_8 A_{11} - 61410960 A_1^2 A_{11}^2 \\
&\quad + 10245847104 A_2^2 A_4 A_{12} - 24350953680 A_4 A_8 A_{17} \\
&\quad - 2913648480 A_4 A_9 A_{17} - 2523363762580 A_1^2 A_5 A_{18}\\
&\quad - 29706323760 A_4 A_8 A_{18} + 334082073870 A_1^2 A_5 A_{19} \\
&\quad +  142776946840 A_1^2 A_5 A_{20} + 47764080 A_4 A_8 A_{20} \\
&\quad + 282210480 A_1^2 A_4 A_{21} + 2047601391150 A_1^2 A_5 A_{21}\\
&\quad + 63016473792 A_2 A_8 A_{22} + 77305513600 A_2 A_9 A_{22} \\
&\quad - 35441430120 A_1^2 A_2 A_{23} - 42056705280 A_2 A_9 A_{23}\\
&\quad - 163762560 A_5^2 A_{29} - 94243374720 A_2 A_5 A_{31}
 + 290822854080 A_2 A_5 A_{32} \\
&\quad - 150861290016 A_2 A_4 A_{34}
  - 47162628000 A_2 A_5 A_{34}),
\end{aligned}\\
\begin{aligned}
\chi_D^{(3)}(\tilde a)
&= (7815 A_2^2 A_4^2 - 1912260 A_4 A_8^2 - 3772362 A_4 A_8 A_9
 - 237900 A_4 A_8 A_{10}\\
&\quad - 178080 A_2 A_{10} A_{13} - 193248 A_2 A_{11} A_{13}
 - 1318176 A_5^2 A_{17} \\
&\quad + 1194740 A_4 A_5 A_{18} - 139104 A_5^2 A_{18} + 56706 A_4 A_5 A_{19} \\
&\quad + 702144 A_5^2 A_{19} - 56552 A_4 A_5 A_{20} - 11040 A_4^2 A_{21}
 - 995070 A_4 A_5 A_{21}\\
&\quad - 32856 A_2 A_4 A_{23} + 26112 A_2 A_5 A_{24}),
\end{aligned}\\
\begin{aligned}
 \chi_E^{(3)}(\tilde a)
&= 54 A_1^2 A_2 + 611 A_2 A_9 - 104 A_2 A_{11} - 140 A_2 A_{12} + 732 A_1 A_{14} \\
&\quad - 243 A_{31} - 234 A_{33} + 245 A_{34},
\end{aligned}\\
\chi_F^{(3)}(\tilde a)= -(11 A_4 + 10 A_5),
\\
\begin{aligned} \\
\chi_A^{(4)}(\tilde a)
&= (-2 A_2^2 A_4 - 80 A_8^2 + 64 A_8 A_9 - 80 A_8 A_{10} + 16 A_9 A_{10}
 - 9 A_{10}^2 - 32 A_8 A_{11} \\
&\quad + 48 A_9 A_{11} + 2 A_{10} A_{11} + 23 A_{11}^2 + 120 A_5 A_{17}
+ 24 A_5 A_{18} - 4 A_5 A_{19}  \\
&\quad + 6 A_4 A_{21} + 4 A_5 A_{21}) (264 A_2^2 A_8 - 112 A_2^2 A_9
 - 56 A_9 A_{17} \\
&\quad + 746 A_{10} A_{17} + 1006 A_{11} A_{17} + 424 A_{10} A_{18}
 + 824 A_{11} A_{18} \\
&\quad + 1092 A_8 A_{19} - 384 A_9 A_{19} - 97 A_{10} A_{19} + 153 A_{11} A_{19}
 - 264 A_8 A_{20}\\
&\quad + 168 A_9 A_{20} + 14 A_{10} A_{20} - 14 A_{11} A_{20} - 620 A_8 A_{21}
+ 81 A_{10} A_{21} \\
&\quad - 81 A_{11} A_{21} + 126 A_4 A_{30} - 208 A_2 A_{31} - 112 A_2 A_{33}),
\end{aligned}\\
\begin{aligned}
\chi_B^{(4)}(\tilde a)
&= (-12 (518 A_8^2 - 16 A_9 (2 A_{10} + 5 A_{11}) +  2 (A_{10}
 + 3 A_{11}) (31 A_{10} + 69 A_{11}) \\
&\quad + A_8 (369 A_{10} + 871 A_{11}) - 96 A_3 A_{17}) + 2 A_5 (552 A_2^2
 - 404 A_{18} + 2271 A_{19} \\
&\quad - 316 A_{20} - 1674 A_{21}) - 135 A_4 A_{21} - 240 A_2 A_{23})
(4 A_2^2 (6160 A_9 \\
&\quad - 60659 A_{10} + 5565 A_{11}) + 533574 A_{10} A_{17}
  + 2120070 A_{11} A_{17} \\
&\quad + 365744 A_{10} A_{18}  + 657528 A_{11} A_{18} - 713634 A_{10} A_{19}
 + 8 A_9 (22484 A_{17} \\
&\quad + 10472 A_{18}  + 10911 A_{19} - 2156 A_{20}) + 121318 A_{10} A_{20}
 - 11130 A_{11} A_{20} \\
&\quad + 522591 A_{10} A_{21} - 357309 A_{11} A_{21} + 72 A_8 (13247 A_{17}
 + 1081 A_{20} \\
&\quad + 7084 A_{21})  + 2079 A_4 A_{30} + 186520 A_2 A_{34}),
\end{aligned} \\
\chi_A^{(5)}(\tilde a)= 95 A_9 + 2 A_{10},\quad
\chi_A^{(6)}(\tilde a)= 4 A_{11} - 4 A_{10},\\
\chi_B^{(6)}(\tilde a)= (A_4 - 2 A_5) ( A_8 - 2 A_{11}),\quad
\chi_A^{(7)}(\tilde a)= (A_3 - A_4) (A_8 - A_{10}),\\
\begin{aligned}
\chi_B^{(7)}(\tilde a)
&= -2 A_8 (6348 A_9^2 - A_4 (502073 A_{18} + 250407 A_{19} + 37072 A_{20}) \\
&\quad + 18 A_2 (720 A_{22} + 8179 A_{23})) + 3 (640 A_9^3
 + 36 A_7^2 (3218 A_{18} + 17721 A_{19}) \\
&\quad + 8 A_9 (7505 A_4 A_{18} + 37966 A_2 A_{23}) + 4 A_2 (A_{13} (74429 A_{18}
  + 44574 A_{19}) \\
&\quad - 7 A_{10} (5387 A_{22} + 4741 A_{23}) + 243552 A_7 A_{27})\\
&\quad  + A_4 (-341504 A_{10} A_{18} - 78779 A_7 A_{25} + 234046 A_2 A_{33})),
\end{aligned}\\
\begin{aligned}
\chi_C^{(7)}(\tilde a)
&= 2484 A_7^2 (2 A_{18} + 9 A_{19}) -
 2 A_8 (276 A_9^2 + A_4 (-34111 A_{18} + 51231 A_{19} \\
&\quad - 35504 A_{20}) +  46794 A_2 A_{23}) + 3 (4 A_2 (5403 A_{13} A_{18}
 - 29222 A_{13} A_{19} \\
&\quad - 6123 A_{10} A_{22} + 11444 A_9 A_{23} + 7131 A_{10} A_{23}\\
&\quad  + 41384 A_7 A_{27}) + A_4 (1080 A_9 A_{18} - 35328 A_{10} A_{18}
- 52173 A_7 A_{25}\\
&\quad + 35842 A_2 A_{33})),
\end{aligned}\\
\chi_D^{(7)}(\tilde a)= (A_3 - A_4) (8 A_7^2 - 44 A_3 A_8 + 27 A_4 A_8
 + 4 A_3 A_9 + 22 A_3 A_{10} - 9 A_4 A_{10}),\\
\chi_F^{(7)}(\tilde a)= 24 A_8 -23 A_{10},\quad
\chi_A^{(8)}(\tilde a)= 5 A_8 - A_9,\\
\chi_D^{(8)}(\tilde a)= 9 A_9 - 25 A_8.
\end{gather*}

We also need here the following additional affine invariant polynomials,
constructed in \cite{SchVul08-JDDE}:
\begin{gather*}
H_2= -\big[(C_1, 8\widehat H+\widehat N)^{(1)}+2D_1\widehat N\big], \quad
H_9= -\big[\widehat D,\widehat D)^{(2)},\widehat D)^{(1)} ,\widehat
 D)^{(3)} \equiv 12 {\rm D},\\
 H_{10}= -\big[\widehat D,\widehat N)^{(2)},D_2)^{(1)},\\
H_{11}= 3\big[(C_1, 8\widehat H+\widehat N)^{(1)}+2D_1\widehat
N\big]^2- 32\widehat H\big[(C_2,\widehat D)^{(2)}+ (\widehat D,D_2)^{(1)}\big], \\
 H_{12}= (\widehat D,\widehat D)^{(2)},\\
 N_7= 12 D_1(C_0,D_2)^{(1)}+2D_1^3+9D_1(C_1,C_2)^{(2)}
+36\big[C_0,C_1)^{(1)},D_2)^{(1)},
\end{gather*}


 Next we construct the following $T$-comitants (for the definition of
$T$-comitants see  \cite{Dana-Vlp-JDE}) which are
 responsible for the existence of invariant straight lines of
 systems \eqref{sys:QSgenCoef}:
\begin{equation} \label{Comit:Bi}
\begin{gathered}
B_3(\tilde a,x,y)=(C_2,\widehat D)^{(1)}=\operatorname{Jacob}( C_2,\widehat D),\\
B_2(\tilde a,x,y)=\left(B_3,B_3\right)^{(2)} - 6B_3(C_2,\widehat D)^{(3)},\\
B_1(\tilde a)=\operatorname{Res}_x\left( C_2,\widehat D\right)/y^9=-2^{-9}3^{-8}\left(B_2,B_3\right)^{(4)}.
\end{gathered}
\end{equation}

\begin{lemma}[see \cite{SchVul04-QTDS}] \label{lem:Bi-0}
For the existence of invariant straight lines in one (respectively 2; 3 distinct)
directions in the affine plane it is necessary that $B_1=0$
(respectively $B_2=0$; $B_3=0$).
\end{lemma}

At the moment we only have necessary and not necessary and
sufficient conditions for the existence of an invariant straight
line or for invariant lines in two or three directions.

Let us apply the translation $x=x'+x_0$, $y=y'+y_0$ to the polynomials
 $p(\tilde{a},x,y)$ and $q(\tilde{a},x,y)$. Then
we obtain $\hat{p}(\hat{a}(a,x_0,y_0),x',y')=p(\tilde{a}, x'+x_0, y'+y_0)$,
$\hat{q}(\hat{a}(a,x_0,y_0),x',y')=q(\tilde{a}, x'+x_0, y'+y_0)$.
Let us construct the following polynomials
\begin{gather*}
\Gamma_i(\tilde{a},x_0,y_0)\equiv\operatorname{Res}_{x'}
\Big(C_i\big(\hat{a}(\tilde{a},x_0,y_0),x',y'\big),
C_0\big(\hat{a}(\tilde{a},x_0,y_0),x',y'\big)\Big)/(y')^{i+1},\\
\Gamma_i(\tilde{a},x_0,y_0)\in\mathbb{R}[\tilde{a},x_0,y_0],\quad i=1,2.
\end{gather*}

We denote
$$
\tilde{\mathcal{E}}_i(\tilde{a},x,y)
=\Gamma_i(\tilde{a},x_0,y_0)\big|_{\{x_0=x,\, y_0=y\}}
\in \mathbb{R}[\tilde{a},x,y]\quad (i=1,2).
$$

\begin{observation}\label{obs:E1,E2}
We note that the polynomials $\tilde{\mathcal{E}}_1(a,x,y)$ and
$\tilde{\mathcal{E}}_2(a,x,y)$ are affine comitants of systems
\eqref{sys:QSgenCoef} and are homogeneous polynomials in the
coefficients $a,b,c,d,e,f,g,h,k,l,m,n$ and non-homogeneous in $x,y$
and $\deg_{\tilde{a}}\tilde{\mathcal{E}}_1=3$,
$\deg_{\,(x,y)}\tilde{\mathcal{E}}_1=5$,
$\deg_{\tilde{a}}\tilde{\mathcal{E}}_2=4$,
$\deg_{\,(x,y)}\tilde{\mathcal{E}}_2=6$.
\end{observation}

Let ${\mathcal{E}}_i(\tilde{a},X,Y,Z)$, $i=1,2$, be the homogenization of
$\tilde{\mathcal{E}}_i(\tilde{a},x,y)$, i.e.
$$
{\mathcal{E}}_1(\tilde{a},X,Y,Z)=Z^5\tilde{\mathcal{E}}_1(\tilde{a},X/Z,Y/Z),\quad
{\mathcal{E}}_2(\tilde{a},X,Y,Z)=Z^6\tilde{\mathcal{E}}_1(\tilde{a},X/Z,Y/Z)
$$
The geometrical meaning of these affine comitants is given by the
following lemma (see \cite{SchVul04-QTDS}):

\begin{lemma}[see \cite{SchVul04-QTDS}] \label{lem:line-L}
(1) The straight line ${\mathcal{L}}(x,y)\equiv ux+vy+w=0$,
$u,v,w\in\mathbb{C}$, $(u,v)\ne(0,0)$ is an invariant line for a quadratic
system \eqref{sys:QSgenCoef} if and only if the polynomial
$\mathcal{L}(x,y)$ is a common factor of the polynomials
$\tilde{\mathcal E}_1(\tilde{a},x,y)$ and $\tilde{\mathcal
E}_2(\tilde{a},x,y)$ over $\mathbb{C}$, i.e.
$$
\tilde{\mathcal E}_i(\tilde{a},x,y)=(ux+vy+w)\widetilde{W}_i(x,y), \quad i=1,2,
$$
where $\widetilde{W}_i(x,y)\in\mathbb{C}[x,y]$.

(2) If $\mathcal{L}(x,y)=0$ is an invariant straight line of multiplicity
$\lambda$ for a quadratic system \eqref{sys:QSgenCoef}, then
$[{\mathcal{L}}(x,y)]^\lambda\mid\gcd(\tilde{\mathcal{E}}_1, \tilde{\mathcal{E}}_2)$
in $\mathbb{C}[x,y]$, i.e. there exist $W_i(\tilde{a},x,y)\in\mathbb{C}[x,y]$, $i=1,2$, such that
$$
\tilde{\mathcal E}_i(\tilde{a},x,y)=(ux+vy+w)^\lambda W_i((a),x,y),\quad i=1,2.
$$

(3) If the line $l_\infty:Z=0$ is of multiplicity $\lambda>1$, then
$Z^{\lambda-1}\mid\gcd({\mathcal E}_1,{\mathcal E}_2)$.
\end{lemma}

To detect the parallel invariant lines we need the
following invariant polynomials:
\begin{gather*}
 N(\tilde a,x,y) = D_2^2 + T_8 - 2 T_9=9\widehat N, \\
 \theta(\tilde a) = 2 A_5 -A_4\ (\equiv \text{Discriminant}\big(N(a,x,y)\big)/1296).
\end{gather*}


\begin{lemma}[see \cite{SchVul04-QTDS}] \label{lem:theta,N}
A necessary condition for the existence of one couple
(respectively two couples) of parallel invariant straight lines of
a system \eqref{sys:QSgenCoef} corresponding to
$\tilde{a}\in\mathbb{R}^{12}$ is the condition $\theta(\tilde{a})=0$
(respectively $N(\tilde{a},x,y)=0$).
\end{lemma}

Now we introduce some important $GL$-comitant in the study of the
invariant conics. Considering
$C_2(\tilde a,x,y)=yp_2(\tilde a,x,y)-xq_2(\tilde a,x,y)$
as a cubic binary form of $x$ and $y$ we calculate
$$
\eta(\tilde a)=\operatorname{Discrim}[C_2/x^3,\xi],\quad
M(\tilde a,x,y)=\operatorname{Hessian}[C_2],
$$
where $\xi=y/x$ or $\xi=x/y$. According to \cite{SchVul10-JFPTA}
we have the next result.


\begin{lemma}[\cite{SchVul10-JFPTA}] \label{lem:S1-S5}
The number of infinite singularities (real and imaginary)
of a quadratic system in {\rm QS}  is determined by the following
conditions:
\begin{itemize}
\item[(i)] $3$ real if $\eta>0$;

\item[(ii)] $1$ real and $2$ imaginary if $\eta<0$;

\item[(iii)] $2$ real if $\eta=0$ and $M\ne0$;

\item[(iv)] $1$ real if $\eta= M=0$ and $C_2\ne0$;

\item[(v)] $\infty$ if $\eta= M= C_2=0$.
\end{itemize}
Moreover, for each one of these cases the quadratic systems
\eqref{sys:QSgenCoef} can be brought via a linear transformation to
one of the following 5 canonical systems:
\begin{gather}
\left\{\begin{gathered}
 \dot x=a+cx+dy+gx^2+(h-1)xy,\\
 \dot y= b+ex+fy+(g-1)xy+hy^2;
\end{gathered} \right.  \label{SI}\\
\left\{\begin{gathered}
 \dot x=a+cx+dy+gx^2+(h+1)xy,\\
 \dot y= b+ex+fy-x^2+gxy+hy^2;
\end{gathered}\right. \label{SII}  \\
\left\{\begin{gathered}
 \dot x=a+cx+dy+gx^2+hxy,\\
 \dot y=b+ex+fy+(g-1)xy+hy^2;
\end{gathered} \right.    \label{SIII} \\
\left\{\begin{gathered}
 \dot x = a+cx+dy+gx^2+hxy,\\
 \dot y = b+ex+fy-x^2+gxy+hy^2;
\end{gathered} \right.     \label{SIV} \\
\left\{\begin{gathered}
 \dot x = a+cx+dy+x^2,\\
 \dot y =  b+ex+fy+xy.
\end{gathered} \right. \label{SV}
\end{gather}
\end{lemma}

Finally,  to detect if an invariant conic
\begin{equation} \label{con:Phi(x,y)}
 \Phi(x,y)\equiv p + q x + r y + s x^2 + 2 t x y + u y^2=0
\end{equation}
(or an invariant line) of a system \eqref{sys:QSgenCoef} has
multiplicity greater than one, we use the notion of
\emph{$k$-th extactic curve $\mathscr{E}_k(X)$} of the vector
field $X$ (see \eqref{vectFild:X}), associated to systems \eqref{sys:QSgenCoef}.
This curve is defined in the paper \cite[Definition 5.1]{ChrLliPer2007}
 by the  polynomial
$$
\mathscr{E}_k(X)=\det\begin{pmatrix}
v_1 & v_2 & \dots & v_l\\
X(v_1) & X(v_2) & \dots & X(v_l)\\
 \dots & \dots & \dots & \dots \\
 X^{l-1}(v_1) & X^{l-1}(v_2) & \dots & X^{l-1}(v_l)
 \end{pmatrix},
$$
where $v_1,v_2,\ldots,v_l$ is the basis of the
$\mathbb{C}$-vector space $\mathbb{C}_n[x,y]$ which is the set of all polynomials in $x,y$
of degree $n$, of polynomials in $\mathbb{C}_n[x,y]$ and
$l=(k+1)(k+2)/2$. Here $ X^0(v_i)=v_i$ and
$X^{j}(v_1)=X(X^{j-1}(v_1))$.
According to \cite{ChrLliPer2007} the following statement holds.

\begin{lemma} \label{lem:Ek}
Assume that an algebraic curve $\Phi(x,y)=0$ of degree $k$ is an
invariant curve for systems \eqref{sys:QSgenCoef}. Then this curve
has multiplicity $m$ if and only if $\Phi(x,y)^m$ divides
$\mathscr{E}_k(X)$.
\end{lemma}


\section{Configurations of invariant hyperbolas for the class QSH$_{(\eta>0)}$}

\begin{theorem}\label{th:Main1}
Consider the class {\rm QSH$_{(\eta>0)}$} of all non-degenerate quadratic
differential systems \eqref{sys:QSgenCoef} possessing three distinct real
singularities at infinity.

{\rm (A)} This family is classified according to the configurations of invariant
hyperbolas
 and of invariant straight lines of the systems, yielding 162 distinct such
configurations.
 This geometric classification appears in  Diagrams \ref{diag:1H-TML1-1} to
  \ref{diag:3TMHge3}. More precisely:
 \begin{itemize}
 \item[(A1)] There are exactly 3 configurations of
 systems possessing an infinite number of hyperbolas.
 \item[(A2)] The remaining 159 configurations could have up to a maximum
of 3 distinct invariant  hyperbolas, real or complex, and up to 4 distinct
invariant straight lines, real or
 complex, including the line at infinity.
 \end{itemize}

{\rm (B)} he bifurcation diagrams for systems in {\rm QSH$_{(\eta>0)}$}
done in the coefficient  space $\mathbb{R}^{12}$ in terms of invariant polynomials
appear in  Diagrams~\ref{diag:eta>0,theta-ne0}
 and  \ref{diag:eta>0,theta=0}. In these diagrams we have necessary and
sufficient conditions  for the realization of each one of the configurations.
\end{theorem}

\begin{figure}[ht]
\centering
\includegraphics[width=0.9\textwidth]{fig3} %1H-TML1-1a.eps
\caption{Configurations with one hyperbola and
$TML=1$}\label{diag:1H-TML1-1}
\end{figure}

\begin{figure}[ht]
\centering
\includegraphics[width=0.9\textwidth]{fig4} % 1H-TML1-2a.eps
\caption{(cont. of Diag. \ref{diag:1H-TML1-1}) 
Configurations with one hyperbola and
$TML=1$} \label{diag:1H-TML1-2}
\end{figure}

\begin{figure}[ht]
 \centering
\includegraphics[width=0.9\textwidth]{fig5} % 1H-TML2-1a.eps
\caption{Configurations with one hyperbola and
$TML=2$}\label{diag:1H-TML2-1}
\end{figure}

\begin{figure}[ht]
 \centering
 \includegraphics[width=0.9\textwidth]{fig6} % 1H-TML2-2a.eps
\caption{(cont. of Diag. \ref{diag:1H-TML2-1})
Configurations with one hyperbola and $TML=2$}
\end{figure}

\begin{figure}[ht]
\centering
 \includegraphics[width=0.9\textwidth]{fig7} % 1H-TML2-3a.eps
\caption{(cont. of Diag. \ref{diag:1H-TML2-1})
Configurations with one hyperbola and
$TML=2$}\label{diag:1H-TML2-3}
\end{figure}


\begin{remark} \label{rmk6}\rm
The invariant polynomials $\chi_W^{(i)}$ in
 Diagrams~\ref{diag:eta>0,theta-ne0} and \ref{diag:eta>0,theta=0},
where $W\in \{A,\ldots,G\}$ and $0\leq i\leq 8$, as well as other
invariant polynomials ($\eta$, $\theta$, $\mu_i$, $\beta_j$\ldots
and so on) are introduced in Section \ref{sec:auxil results}.
Moreover, in these diagrams we denote by $(\mathfrak{C}_i)$ $(i=1,2,3)$
the following sets of conditions
\begin{itemize}
\item[(C1)] $ (\beta_2\mathcal{R}_1\ne0) \cup
(\beta_2=\gamma_3=0\cap  \beta_3\ne0)$,

\item[(C2)] $ (\beta_4\beta_5\mathcal{R}_2\ne0) \cup (\beta_4=\gamma_3=0, \
 \mathcal{R}_2\ne0)$,

\item[(C3)] $(\beta_1=0)\cap\big((\gamma_{12}=0,\mathcal{R}_9\ne0)\cup
(\gamma_{13}=0)\big)\big)$.
\end{itemize}
\end{remark}

\begin{remark} \label{rm:details-Main1} \rm
For more details about the geometric classification of the configurations
of systems in {\rm QSH$_{(\eta>0)}$} see Section \ref{sec:conclusion}.
\end{remark}


\begin{figure}[ht]
\centering
 \includegraphics[width=0.9\textwidth]{fig8} % 1H-TML3-1a.eps
\caption{Configurations with one hyperbola and
$TML=3$}\label{diag:1H-TML3-1}
\end{figure}

\begin{figure}[ht]
\centering
 \includegraphics[width=0.9\textwidth]{fig9} % 1H-TML3-2a.EPS
\caption{(cont. of Diag.\ref{diag:1H-TML3-1}) 
Configurations with one hyperbola and $TML=3$}
\end{figure}

\begin{figure}[ht]
 \centering
 \includegraphics[width=0.9\textwidth]{fig10} % 1H-TMLge4a.eps
\caption{Configurations with one hyperbola and
$TML\ge4$}\label{diag:1H-TMLge4}
\end{figure}

\begin{figure}[ht]
 \centering
 \includegraphics[width=0.9\textwidth]{fig11} % 2H-TML1a.eps
\caption{Diagram of configurations with two hyperbolas and
$TML=1$}\label{diag:2H-TML1}
\end{figure}

\begin{figure}[ht]
\centering
 \includegraphics[width=0.9\textwidth]{fig12} % 2H-TMLge2-1a.eps
\caption{Diagram of configurations with two hyperbolas and
$TML\ge2$}\label{diag:2H-TMLge2-1}
\end{figure}

\begin{figure}[ht]
 \centering
\includegraphics[width=0.9\textwidth]{fig13} % 2H-TMLge2-2a.eps
\caption{(cont. of Diag.\ref{diag:2H-TMLge2-1})
Diagram of configurations with two hyperbolas and $TML\ge2$}
\end{figure}

\begin{figure}[ht]
\centering
 \includegraphics[width=0.9\textwidth]{fig14} % 3TMHge3a.eps
\caption{Diagram of configurations with three or more
hyperbolas ($TMH\ge3$)}\label{diag:3TMHge3}
\end{figure}

\begin{figure}[ht]
 \centering
\includegraphics[width=0.8\textwidth]{fig15}
  % Diagram-Hyp-IL-eta-pos-theta-ne-0-1b.eps
\caption{Bifurcation diagram in
$\mathbb{R}^{12}$ of the configurations: Case $\eta>0$,
$\theta\ne0$}\label{diag:eta>0,theta-ne0}
\end{figure}

\begin{figure}[ht]
\centering
 \includegraphics[width=0.75\textwidth]{fig16} 
 %Diagram-Hyp-IL-eta-pos-theta-ne-0-2b.eps
\caption{(cont. Diag. \ref{diag:eta>0,theta-ne0})
Bifurcation diagram in $\mathbb{R}^{12}$ of the
configurations: Case $\eta>0$, $\theta\ne0$ }
\end{figure}

\begin{figure}[ht]
 \centering
\includegraphics[width=0.9\textwidth]{fig17}
 % Diagram-Hyp-IL-eta-pos-theta-ne-0-3b.eps
\caption{(cont. of Diag. \ref{diag:eta>0,theta-ne0})
Bifurcation diagram in $\mathbb{R}^{12}$ of the
configurations: Case $\eta>0$, $\theta\ne0$}
\end{figure}


 \begin{figure}[ht]
\centering
 \includegraphics[width=0.8\textwidth]{fig18} 
 %Diagram-Hyp-IL-eta-pos-theta-ne-0-4b.eps
\caption{(cont. of Diag.\ref{diag:eta>0,theta-ne0})
Bifurcation diagram in $\mathbb{R}^{12}$ of the
configurations: Case $\eta>0$, $\theta\ne0$}
\end{figure}

\begin{figure}[ht]
\centering
 \includegraphics[width=0.9\textwidth]{fig19} 
% Diagram-Hyp-IL-eta-pos-theta-ne-0-5b.eps
\caption{(cont. of Diag. \ref{diag:eta>0,theta-ne0})
Bifurcation diagram in $\mathbb{R}^{12}$ of the
configurations: Case $\eta>0$, $\theta\ne0$}
\end{figure}


\begin{proof}[Proof of Theorem \ref{th:Main1}]
We prove part (A) under the assumption that part
(B) is already proved. Later, we prove part (B).

We first need to make sure that the concepts introduced
above gave us a sufficient number of invariants under the action
of the affine group and time rescaling so as to be able to
classify geometrically the class QSH$_{(\eta>0)}$ according to their
configurations of their invariant hyperbolas and lines. Summing up
all the concepts introduced, we end up with the list: (CD,
MS$_{0C}$, TMH, TML, PD, PD$_f$, PD$_\infty$, mP,
I. From this list we clearly have that TMH and TML are
invariants under the group action because the action conserves
lines and the type of a conic as well as parallelism and it
conserves singularities of the systems which are simple points on
an invariant curve. The types of the divisor (CD on $P_2(\mathbb{C})$
and of the zero-cycle MS$_{0C}$ on $P_2(\mathbb{R})$ are invariants
under the group because the group conserves the multiplicities of
the invariant curves as well as the multiplicities of the
singularities. The number of vertices of a basic polygon is
conserved under the group action basically because the number of
intersection points of the various invariant curves is conserved.
Furthermore the coefficients of mP are also conserved because
multiplicities of the singularities are conserved. For analogous
reasons the coefficients of PD, PD$_f$, PD$_\infty$ are also
conserved. The invariant I is also conserved because complex
intersection points of a line with a hyperbola as well as
intersection multiplicities are conserved. The concepts involved
above yield all the invariants we need and we now prove that the
162 configurations obtained in this section
are distinct.

Fixing the values of TMH and TML, we first apply the main
divisor (CD. In many cases, just using the invariants contained
in (CD and the zero-cycle MS$_{0C}$ (TMH, TML and the
corresponding types) suffice for distinguishing the
configurations in a group of configurations. In other cases more
invariants are needed and we introduce the necessary additional
invariants, to distinguish the configurations of the following
groups. The result is seen in the Diagrams~\ref{diag:1H-TML1-1} to
\ref{diag:3TMHge3}.

We finally obtain that the 162 geometric configurations displayed in
 Diagrams~\ref{diag:1H-TML1-1} to \ref{diag:3TMHge3} are distinct,
which yields the geometric classification of the class QSH
according to the configurations of invariant hyperbolas and lines.
This proves statement (A) of this theorem.

\begin{figure}[ht]
 \centering
 \includegraphics[width=0.9\textwidth]{fig20}
 % Diagram-Hyp-IL-eta-pos-theta0-1b.eps 
\caption{Bifurcation diagram in
$\mathbb{R}^{12}$ of the configurations: Case $\eta>0$,
$\theta=0$}\label{diag:eta>0,theta=0}
\end{figure}

\begin{figure}[ht]
\centering
 \includegraphics[width=0.9\textwidth]{fig21} 
% Diagram-Hyp-IL-eta-pos-theta0-2b.eps
\caption{(cont. of Diag.\ref{diag:eta>0,theta=0})
Bifurcation diagram in $\mathbb{R}^{12}$ of the
configurations: Case $\eta>0$, $\theta=0$}
\end{figure}


\begin{figure}[ht]
\centering
 \includegraphics[width=0.9\textwidth]{fig22}
 % Diagram-Hyp-IL-eta-pos-theta0-3b.eps
\caption{(cont. of Diag. \ref{diag:eta>0,theta=0})
Bifurcation diagram in $\mathbb{R}^{12}$ of the
configurations: Case $\eta>0$, $\theta=0$}
\end{figure}

\begin{figure}[ht]
 \centering
 \includegraphics[width=0.9\textwidth]{fig23}
 % Diagram-Hyp-IL-eta-pos-theta0-4b.eps
\caption{(cont. of Diag. \ref{diag:eta>0,theta=0})
Bifurcation diagram in $\mathbb{R}^{12}$ of the
configurations: Case $\eta>0$, $\theta=0$}
\end{figure}



\noindent\emph{Proof of part} (B).
We assume $\eta>0$. In this case according to
\cite[Lemma 44]{SchVul04-QTDS} there exist an affine
transformation and time rescaling which brings systems
\eqref{sys:QSgenCoef} to the systems
\begin{equation} \label{sys:eta-poz-Gen}
\begin{gathered}
 \frac {dx}{dt}= a+cx+dy+gx^2+(h-1)xy,\quad
 \frac {dy}{dt}= b+ex+fy+(g-1)xy+hy^2,
\end{gathered}
\end{equation}
with $\eta=1$ and $\theta=- (g-1) (h-1) (g + h)/2$.
\end{proof}

\subsection{Subcase $\theta\ne0$}
Following Theorem \ref{trm:exist-hyp} we assume that for a
quadratic system
\eqref{sys:eta-poz-Gen} the conditions $\theta\ne0$ and
$\gamma_1=0$ are fulfilled. Then, as it was proved in
\cite{Oliv-Rez-Vulpe}, by an affine transformation and time
rescaling, this system could be brought to the canonical form
\begin{equation} \label{sys:eta-poz-theta-ne0-gamma1=0}
 \frac {dx}{dt}= a+cx+gx^2+(h-1)xy,\quad
 \frac {dy}{dt}= b-cy+(g-1)xy+hy^2,
\end{equation}
for which we calculate
\begin{equation} \label{val:gamma1=0}
\begin{gathered}
\gamma_2=-1575 c^2 (g-1)^2 (h-1)^2 (g+h)(3g-1) (3h-1) (3g+3h-4) \mathcal{B}_1,\\
\beta_1=- c^2 (g-1) (h-1) (3g-1) (3h-1)/4,\\
\beta_2=- c (g-h) (3g+3h-4)/2,\quad \theta=- (g-1) (h-1) (g+h)/2,
\end{gathered}
\end{equation}
where $\mathcal{B}_1=b(2h-1)-a(2g-1)$.

\subsubsection{Possibility $\beta_1\ne0$}\label{subsec:CS-1}
In this case the condition $\gamma_2=0$ is equivalent to
 $(3g+3h-4)\mathcal{B}_1=0$.

\paragraph{Case $\beta_2\ne0$.}
Then $3g+3h-4\ne0$ and we obtain
$\mathcal{B}_1=0$. Since $c\ne0$ from the rescaling
$(x,y,t)\mapsto (cx,cy,t/c)$ we may assume $c=1$. Moreover as
$(2g-1)^2+(2h-1)^2\ne0$ because $\beta_2\ne0$ (i.e. $g-h\ne0$), the
condition $\mathcal{B}_1=0$ could be written as $a=a_1(2h-1)$ and
$b=a_1(2g-1)$. So setting the old parameter $a$ instead of $a_1$,
we arrive at the 3-parameter family of systems
\begin{equation} \label{sys:CS-1}
\begin{gathered}
 \frac {dx}{dt}= a(2h-1)+ x+gx^2+(h-1)xy,\quad
 \frac {dy}{dt}= a(2g-1)- y+(g-1)xy+hy^2
\end{gathered}
\end{equation}
with the condition
\begin{equation} \label{cond:CS-1}
a (g-1)(h-1)(g + h)(g - h) (3g-1)(3h-1)(3 g + 3 h-4) \ne0.
\end{equation}
These systems possess the invariant hyperbola
\begin{equation} \label{Hyp:1}
\Phi(x,y)=a+xy=0.
\end{equation}

\begin{remark} \label{rem:g<->h} \rm
We point out that for systems \eqref{sys:CS-1} the parameters $g$
and $h$ have the same significance, because we could replace $g$
by $h$ via the change $(x,y,t,a,g,h)\mapsto (-y,-x,-t,a,h,g)$,
which brings a system to one of the same form.
\end{remark}

For systems \eqref{sys:CS-1} we calculate
\begin{equation} \label{val:B1-CS1}
B_1= 2 a^2 (g-1)^2 (h-1)^2 (g-h) (2g-1) (2h-1) \big[a(g+h)^2-1\big].
\end{equation}

\subparagraph{Subcase $B_1\ne0$.} In this case by Lemma
\ref{lem:Bi-0} we have no invariant lines. For systems
\eqref{sys:CS-1} we calculate\ $ \mu_0=g h (g + h-1) $ and we
consider two possibilities: $\mu_0\ne0$ and $\mu_0=0$.


(a) {\it Possibility $\mu_0\ne0$.} Then by Lemma
\ref{lem:mu_i-ISPs} the systems have finite singularities of
total multiplicity 4. We detect that two of these singularities
are located on the hyperbola, more exactly such singularities are
$M_{1,2}\big(x_{1,2} ,y_{1,2} \big)$ with
\begin{equation} \label{page:x_i,y_i}
x_{1,2}=\frac{-1 \pm \sqrt{Z_1}}{2g},\quad
y_{1,2}= \frac{1 \pm \sqrt{Z_1}}{2h},\quad
Z_1=1-4agh.
\end{equation}
On the other hand for systems \eqref{sys:CS-1} we calculate the
invariant polynomial
\begin{gather*}
\chi_A^{(1)}=(g-1)^2 (h-1)^2 (g-h)^2 (3g-1)^2 (3h-1)^2\,Z_1,\\
\chi_B^{(1)}=-105a (g-1)^2 (h-1)^2 (g-h)^2 (3g-1)^2 (3h-1)^2/8
\end{gather*}
and by \eqref{cond:CS-1} we conclude that 
$\operatorname{sign}(\chi_A^{(1)})=\operatorname{sign}(Z_1)$
(if $Z_1\ne 0$) and $\operatorname{sign}(\chi_B^{(1)})=-\operatorname{sign}(a)$.
So we consider three cases: $\chi_A^{(1)}<0$, $\chi_A^{(1)}>0$ and
$\chi_A^{(1)}=0$.

(a1) {\it Case $\chi_A^{(1)}<0$.} So we have
no real singularities located on the invariant hyperbola and we
arrive at the configurations of invariant curves given by
Config. H.1 if $\chi_B^{(1)}<0$ and
Config. H.2 if $\chi_B^{(1)}>0$.

(a2)  {\it Case $\chi_A^{(1)}>0$.} In this case we
have two real singularities located on the hyperbola. We have the
next result.

\begin{lemma} \label{lem:sign(x1*x2)b}
Assume that the singularities $M_{1,2}\big(x_{1,2} ,y_{1,2} \big)$
(located on the hyperbola) are finite. Then these singularities
are located on different branches of the hyperbola if $\chi_C^{(1)}<0$
and they are located on the same branch if $\chi_C^{(1)}>0$.
\end{lemma}

\begin{proof}
Since the asymptotes of the hyperbola \eqref{Hyp:1} are
the lines $x=0$ and $y=0$ it is clear that the singularities
$M_{1,2}$ are located on different branches of the hyperbola if and
only if $x_1x_2<0$. We calculate
\begin{equation} \label{val:chi2}
\begin{gathered}
x_1x_2=\big[\frac{-1+\sqrt{Z_1}}{2g}\big] \big[\frac{-1-\sqrt{Z_1}}{2g}\big]
=\frac{ah}{g},\\
\chi_C^{(1)}=35agh (g-1)^4 (h-1)^4 (g-h)^2 (g+h)^2 (3g-1)^2 (3h-1)^2/32\,.
\end{gathered}
\end{equation}
By condition \eqref{cond:CS-1} we obtain that
$\operatorname{sign}(x_1x_2)=\operatorname{sign}(\chi_C^{(1)})$. This completes the proof of the
lemma.
\end{proof}

Other two singular points of systems \eqref{sys:CS-1}are
$M_{3,4}\big(x_{3,4} ,y_{3,4} \big)$ (generically located outside the
hyperbola) with
\begin{equation} \label{coord:M3,M4-CS1}
\begin{gathered}
x_{3,4}=\frac{(1-2h)\big[1\pm\sqrt{Z_2}\big]}{2(g+h-1)},\quad
y_{3,4}=\frac{(2g-1)\big[1\pm\sqrt{Z_2}\big]}{2(g+h-1)},\\
 Z_2=1+4a(1-g-h).
\end{gathered}
\end{equation}

We need to determine the conditions when the singular points
located outside the hyperbola coincide with its points (singular
for the systems or not). In this order considering \eqref{Hyp:1}
we calculate
$$
\Phi(x,y)|_{\{x=x_{3,4},\ y=y_{3,4}\}}
= \frac{\tilde{A}-(2g-1)(2h-1)\big[1\pm\sqrt{Z_2}\big]}{2(g+h-1)^2}
\equiv \Omega_{3,4}(a,g,h),
$$
where $\tilde{A}=2a(g+h-1)(4gh-g-h)$. It is clear that at least one
of the singular points $M_3(x_3,y_3)$ or $M_4(x_4,y_4)$ belongs
to the hyperbola \eqref{Hyp:1} if and only if
$$
\Omega_3 \Omega_4= -\frac{a\,Z_3}{(g+h-1)^2}=0,\quad
 Z_3=(2g-1)(2h-1)-a(4gh-g-h)^2.
$$
On the other hand for systems \eqref{sys:CS-1} we have
$$
\chi_D^{(1)}=105(g-h) (3g-1) (3h-1)\,Z_3/4
$$
and clearly by \eqref{cond:CS-1} the condition $\chi_D^{(1)}=0$ is
equivalent to $Z_3=0$. We examine two subcases: $\chi_D^{(1)}\ne0$
and $\chi_D^{(1)}=0$.


($\alpha$) {\it Subcase $\chi_D^{(1)}\ne0$.} Then
$Z_3\ne0$ and on the hyperbola there are two simple real
singularities (namely $M_{1,2} (x_{1,2} ,y_{1,2})$). By Lemma
\ref{lem:sign(x1*x2)} their position is defined by the invariant
polynomial $\chi_C^{(1)}$ and we arrive at the following
conditions and configurations:
\begin{itemize}
 \item $\chi_C^{(1)}<0$ and $\chi_B^{(1)}<0$   $\Rightarrow$  Config. H.17;
 \item $\chi_C^{(1)}<0$ and $\chi_B^{(1)}>0$   $\Rightarrow$  Config. H.19;
 \item $\chi_C^{(1)}>0$ and $\chi_B^{(1)}<0$   $\Rightarrow$  Config. H.16;
 \item $\chi_C^{(1)}>0$ and $\chi_B^{(1)}>0$   $\Rightarrow$  Config. H.18.
\end{itemize}


($\beta$) {\it Subcase $\chi_D^{(1)}=0$.} In this case
the conditions $Z_3=0$, $B_1\ne0$ (see \eqref{val:B1-CS1}) and
\eqref{cond:CS-1} implies $4gh-g-h\ne0$ and we obtain $a=(2g-1)(2h-1)/(4gh-g-h)^2$.
Then considering Proposition~\ref{prop:number-FSPs} we calculate
\begin{gather*}
{\rm D}=0,\quad {\rm T}=-3\big[g(g-1)(2h-1)x+h(h-1)(2g-1)y\big]^2{\rm P},\\
{\rm P}= \frac{(g-h)^2}{(4gh-g-h)^4}(2-3g-3h+4gh)^2 (gx-hy)^2
\big[(2g-1)x+(2h-1)y\big]^2.
\end{gather*}


($\beta1$) {\it Possibility ${\rm T}\ne0$.} Then
${\rm T}<0$ and according to Proposition \ref{prop:number-FSPs}
systems \eqref{sys:CS-1} possess one double and two simple real
finite singularities. More exactly, we detect that one of the singular
 points $M_3(x_3,y_3)$ or $M_4(x_4,y_4)$ coalesced with a singular point
located on the hyperbola, whereas another one remains outside the hyperbola.
Taking into consideration Lemma \ref{lem:sign(x1*x2)} we obtain the
following conditions and configurations:
\begin{itemize}
 \item $\chi_C^{(1)}<0$ and $\chi_B^{(1)}<0$  $\Rightarrow$ Config. H.21;
 \item $\chi_C^{(1)}<0$ and $\chi_B^{(1)}>0$  $\Rightarrow$ Config. H.23;
 \item $\chi_C^{(1)}>0$ and $\chi_B^{(1)}<0$  $\Rightarrow$ Config. H.20;
 \item $\chi_C^{(1)}>0$ and $\chi_B^{(1)}>0$  $\Rightarrow$ Config. H.22.
\end{itemize}


($\beta2$) {\it Possibility ${\rm T}=0$.} In this case
by  conditions \eqref{cond:CS-1} and $\mu_0\ne0$ the
equality ${\rm T}=0$ holds if and only if ${\rm P}=0$ which is equivalent
to $2-3g-3h+4gh=0$ (or equivalently $2-3g+h(4g-3)=0$). Since
$g-h\ne0$ (see \eqref{cond:CS-1}), the condition
$(4g-3)^2+(4h-3)^2\ne0$ holds, then by Remark \ref{rem:g<->h} we
may assume $(4g-3)\ne0$, i.e. $h=(3g-2)/(4g-3)$ and we obtain
$$
 {\rm D}={\rm T}={\rm P}=0,\quad {\rm R}
=\frac{3}{(4g-3)^4}(g-1)^2(2g-1)^2 \big[g(4g-3)x+(2-3g)y\big]^2.
$$
Since ${\rm R}\ne0$, by Proposition \ref{prop:number-FSPs} we obtain one
triple and one simple singularities. More precisely the singular points
$M_3$ and $M_4$ coalesced with one of the singular points $M_1$ or $M_2$
and the last point becomes a triple one. In this case, we calculate
\begin{gather*}
\chi_B^{(1)}=-\frac{105 (g-1)^6 (3g-1)^2 (5g-3)^2}{8(4g-3)^5}, \\
\chi_C^{(1)}=\frac{35\,g(3g-2) (g-1)^{10} (3g-1)^2 (5g-3)^2
(2g^2-1)^2}{8(4g-3)^{10}}.
\end{gather*}
We remark that the condition $\chi_C^{(1)}<0$ implies $\chi_B^{(1)}>0$.
Indeed, if $\chi_C^{(1)}<0$ then $g(3g-2)<0$ (i.e. $0<g<2/3$) and for
these values of $g$ we have $4g-3<0$, which is equivalent to $\chi_B^{(1)}>0$.
 Taking into consideration Lemma~\ref{lem:sign(x1*x2)} we obtain the
following conditions and configurations:
\begin{itemize}
 \item $\chi_C^{(1)}<0$   $\Rightarrow$  Config. H.26;
 \item $\chi_C^{(1)}>0$ and $\chi_B^{(1)}<0$  $\Rightarrow$ Config. H.24;
 \item $\chi_C^{(1)}>0$ and $\chi_B^{(1)}>0$  $\Rightarrow$ Config. H.25.
\end{itemize}


(a3)  {\it Case $\chi_A^{(1)}=0$.} By condition \eqref{cond:CS-1},
the condition $\chi_A^{(1)}=0$ implies $Z_1=0$ and it yields $a=1/(4gh)$.
In this case the points $M_{1,2}$ coalesce and we have a double point on
the hyperbola. So we calculate
\begin{gather*}
 \chi_B^{(1)}=-\frac{105}{32gh}(g-1)^2(h-1)^2(g-h)^2 (3g-1)^2(3h-1)^2,\\
 \chi_C^{(1)}=35(g-1)^4 (h-1)^4 (g-h)^2 (g+h)^2 (3g-1)^2 (3h-1)^2/128>0,\\
 \chi_D^{(1)}=-\frac{105}{16gh} (g-h)^3 (3g-1) (3h-1)\ne0.
\end{gather*}
Since $\chi_C^{(1)}\ne0$, no other point could coalesce with the
double point on the hyperbola and we arrive at the configurations
given by Config. H.7 if $\chi_B^{(1)}<0$ and
Config. H.8 if $\chi_B^{(1)}>0$.


(b) {\it Possibility $\mu_0=0$.} Then by Lemma \ref{lem:mu_i-ISPs}
at least one finite singular point has gone to
infinity and collapsed with one of the infinite singular points
$[1,0,0]$, $[0,1,0]$ or $[1,1,0]$. By the same lemma, a second
point could go to infinity if and only if $\mu_1(x,y)=0$. However,
for systems \eqref{cond:CS-1} we have the following remark.

\begin{remark} \label{rem:mu0=0,mu1-ne0} \rm
If for a system \eqref{sys:CS-1} the condition $\mu_0=0$ holds
then $\mu_1\ne0$. Moreover by \eqref{Pol:D,T,R,e.al.} the
condition ${\rm R}=3\mu_1^2\ne0$ is fulfilled.
\end{remark}

 Indeed for systems \eqref{sys:CS-1} we calculate
\begin{equation} \label{val:mu0,mu1-CS1}
\mu_0= g h (g + h-1)=0,\quad \mu_1=
 g(1-g-2gh)x+h(1-h-2gh)y.
\end{equation}
We observe that in the case $g=0$ (respectively $h=0$; $g=1-h$) we
get $\mu_1= h(1-h)y\ne0$ (respectively $\mu_1= g(1-g)y\ne0$;
$\mu_1= h(h-1)(2h-1)(x-y)\ne0$) by condition
\eqref{cond:CS-1}.

We consider the cases: $\chi_A^{(1)}<0$, $\chi_A^{(1)}>0$ and $\chi_A^{(1)}=0$.


(b1) {\it Case $\chi_A^{(1)}<0$.} The points
on the hyperbola are complex and, moreover, $1-4agh<0$ implies
$agh>0$ and hence $\chi_C^{(1)}>0$. Then we arrive at the
configurations given by Config. H.3 if $\chi_B^{(1)}<0$,
and Config. H.4 if $\chi_B^{(1)}>0$.


(b2) {\it Case $\chi_A^{(1)}>0$.}
The points on the hyperbola are real and we observe that because of
condition \eqref{cond:CS-1} the equality $\chi_C^{(1)}=0$ is
 equivalent to $gh=0$. So we consider two subcases:
 $\chi_C^{(1)}\ne0$ and $\chi_C^{(1)}=0$.

($\alpha)$ {\it Subcase $\chi_C^{(1)}\ne0$.} Then the
condition $\mu_0=0$ gives $g+h-1=0$, i.e. $g=1-h$ and one finite
singularity has gone to infinity and collapsed with the point
$[1,1,0]$. Clearly that this must be a singular point located
outside the hyperbola and hence on the finite part of the phase
plane of systems \eqref{sys:CS-1} there are three singularities,
two of which ($M_1$ and $M_2$) being located on the hyperbola.

Since the singular points on the hyperbola are real we have to decide
 when the third point will belong also to the hyperbola.
For systems \eqref{sys:CS-1} with $g=1-h$ we calculate
\begin{gather*}
\chi_B^{(1)}=-105 a h^2 (h-1)^2 (2h-1)^2 (3h-1)^2 (3h-2)^2/8,\\
\chi_D^{(1)}= 105 (2h-1)^3 (2-3h) (3h-1) \big[1+a(2h-1)^2\big]/4.
\end{gather*}

We observe that the condition $\chi_B^{(1)}<0$ implies $\chi_D^{(1)}\ne0$.
Indeed, supposing $\chi_D^{(1)}=0$ and considering condition \eqref{cond:CS-1},
 we obtain $a=-1/(2h-1)^2$ and hence
$$
\chi_B^{(1)}=105 h^2 (h-1)^2 (3h-1)^2 (3h-2)^2/8>0.
$$

So in the case $\chi_C^{(1)}<0$ we get the following conditions and configurations:
\begin{itemize}
 \item $\chi_B^{(1)}<0$  $\Rightarrow$ Config. H.30;
 \item $\chi_B^{(1)}>0$ and $\chi_D^{(1)}\ne0$ $\Rightarrow$ Config. H.32;
 \item $\chi_B^{(1)}>0$ and $\chi_D^{(1)}=0$  $\Rightarrow$  Config. H.34;
\end{itemize}
whereas for $\chi_C^{(1)}>0$ we get
\begin{itemize}
 \item $\chi_B^{(1)}<0$  $\Rightarrow$  Config. H.29;
 \item $\chi_B^{(1)}>0$ and $\chi_D^{(1)}\ne0$  $\Rightarrow$  Config. H.31;
 \item $\chi_B^{(1)}>0$ and $\chi_D^{(1)}=0$  $\Rightarrow$ Config. H.33.
\end{itemize}


($\beta$) {\it Subcase $\chi_C^{(1)}=0$.} Then $gh=0$ and
$g^2+h^2\ne0$ because $g-h\ne0$. By Remark \ref{rem:g<->h} we may
assume $g=0$ and then one of the singularities located on the
hyperbola \eqref{Hyp:1} has gone to infinity and collapsed with
the point $[1,0,0]$. The calculations yield
\begin{equation} \label{val:Chi2,Chi4}
\begin{gathered}
 \chi_B^{(1)}=-105 a h^2 (h-1)^2 (3h-1)^2/8,\quad
 \chi_D^{(1)}=105 h (3h-1) (1-2h-ah^2)/4.
 \end{gathered}
\end{equation}


($\beta1$) {\it Possibility $\chi_B^{(1)}<0$.}
Then we have to analyze two cases: $\chi_D^{(1)}\ne0$ and $\chi_D^{(1)}=0$.

If $\chi_D^{(1)}\ne0$, the finite singularities $M_{3,4}$ remain
outside the hyperbola and we arrive at the configuration given by
Config. H.9. In the case $\chi_D^{(1)}=0$ (which yields
$a=(1-2h)/h^2$), one of the singular points $M_{3,4}$ coalesces
with the remaining singularity on the hyperbola. For this case we
calculate
$$
{\rm D}=0, \quad {\rm P}= (3h-2)^2 y^2 (x+y-2hy)^2, \quad
{\rm T}=-3h^2(h-1)^2y^2{\rm P}.
$$
We observe that the condition $\chi_B^{(1)}>0$ implies ${\rm T}\ne0$.
Indeed, the conditions $\chi_D^{(1)}={\rm T}=0$ imply $h=2/3$ and $a=-3/4$,
and hence $\chi_B^{(1)}>0$.

Moreover, according to Remark \ref{rem:mu0=0,mu1-ne0}, in the case
$\mu_0=0$, the condition ${\rm R}\ne0$ is satisfied for systems
\eqref{sys:CS-1}. Then, since ${\rm T}\ne0$, we obtain ${\rm P}{\rm R}\ne0$,
and by Proposition~\ref{prop:number-FSPs} we have a double
singular point on the hyperbola and we arrive at
Config. H.11.


($\beta2$) {\it Possibility $\chi_B^{(1)}>0$.} We
again analyze the cases $\chi_D^{(1)}\ne0$ and $\chi_D^{(1)}=0$.
In the case $\chi_D^{(1)}\ne0$, the finite singularities $M_{3,4}$
remain outside the hyperbola and we arrive at the configuration
given by Config. H.10. If $\chi_D^{(1)}=0$, we obtain
the configurations shown in Config. H.12 if ${\rm T}\ne0$,
and Config. H.13 if ${\rm T}=0$.


(b3) {\it Case $\chi_A^{(1)}=0$.} By condition \eqref{cond:CS-1},
the condition $\chi_A^{(1)}=0$ implies $Z_1=0$ (then $gh\ne0$) and
hence $a=1/(4gh)$. Therefore the condition $\mu_0=0$ yields $g=1-h$.
In this case the singular points $M_{1,2}$ coalesce and we have a
double point on the hyperbola. For systems \eqref{sys:CS-1} with $g=1-h$
and $a=1/[4h(1-h)]$, we calculate
\begin{gather*}
 \chi_B^{(1)}=105 h (h-1) (2h-1)^2 (3h-1)^2 (3h-2)^2/32,\\
 \chi_C^{(1)}=\frac{35}{128}\,h^4 (h-1)^4 (2h-1)^2 (3h-1)^2 (3h-2)^2,\\
 \chi_D^{(1)}=\frac{105}{16h(h-1)}\,(2h-1)^3 (3h-1) (3h-2),\\
 {\rm D}=0,\quad {\rm T}=-3 h^2 (h-1)^2 (2h-1)^2 (x-y)^4 (x+y)^2\ne0.
\end{gather*}
Since $\chi_D^{(1)}\ne0$ (from condition \eqref{cond:CS-1}), the
singular point located outside the hyperbola could not collapse
with this double point and we arrive at the configurations given
by Config. H.14 if $\chi_B^{(1)}<0$ and
Config. H.15 if $\chi_B^{(1)}>0$.

\subparagraph{Subcase $B_1=0$.} According to Lemma \ref{lem:Bi-0}
the condition $B_1=0$ is necessary in order to exist an
invariant line of systems \eqref{sys:CS-1}. Considering the
condition \eqref{cond:CS-1} we obtain that $B_1=0$ (see
\eqref{val:B1-CS1}) is equivalent to
$$
(2g-1)(2h-1)\big[a(g+h)^2-1\big]=0.
$$
On the other hand, for these systems we calculate
$$
\chi_E^{(1)}=-105(g-1)(h-1)(g-h)(3g-1)(3h-1)\,Z_4, \quad
 Z_4=\big[a(g+h)^2-1\big],
$$
and by \eqref{cond:CS-1} the condition $Z_4=0$ is equivalent
to $\chi_E^{(1)}=0$.


(a) {\it Possibility $\chi_E^{(1)}\ne0$.} In this case we
get $g=1/2$ and this leads to the systems
\begin{equation} \label{sys:CS-1-IL}
\frac {dx}{dt}= a(2h-1)+ x+ x^2/2+(h-1)xy,\quad
 \frac {dy}{dt}= - y(2+x-2hy)/2,
\end{equation}
for which the following condition holds (see \eqref{cond:CS-1}):
\begin{equation} \label{cond:CS-1-IL}
a (h-1)(2h-1)(2h+1)(3h-1)(6h -5) \ne0.
\end{equation}

 We observe that besides the hyperbola
\eqref{Hyp:1} these systems possess the invariant line $y=0$,
which is one of the asymptotes of this hyperbola. For the above
systems we calculate
\begin{gather*}
\mu_0=h (2h-1)/4,\quad \chi_E^{(1)}=-\frac{105}{8}(h-1)(2h-1)(3h-1)Z_4
\big|_{\{g=1/2\}},\\
B_1=0,\quad  B_2=-648 a (h-1)^2 (2h-1)^2 y^4\,Z_4\big|_{\{g=1/2\}}.
\end{gather*}
Therefore we conclude that by conditions $\chi_E^{(1)}\ne0$ and
\eqref{cond:CS-1-IL} we obtain $B_2\ne0$ and, by Lemma~\ref{lem:Bi-0},
 we could not have an invariant line in a direction which is different
from $y=0$. Moreover, by condition $\theta\ne0$ and according to
Lemma~\ref{lem:theta,N}, in the direction $y=0$ we could not have either
 a couple of parallel invariant lines or a double invariant line.


(a1) {\it Case $\mu_0\ne0$.} Then $h(2h-1)\ne0$ and considering the
coordinates of the singularities $M_i(x_i,y_i)$ ($i=1,2,3,4$) mentioned
earlier (see page \pageref{page:x_i,y_i}) for $g=1/2$ we have
\begin{gather*}
x_{1,2}=-1\pm\sqrt{1-2ah},\quad
y_{1,2}=-1\mp\sqrt{1-2ah},\\
x_{3,4}=-1\pm\sqrt{1+2a(1-2h)},\quad y_{3,4}= 0.
\end{gather*}
We recall that the singular points $M_{1,2}(x_{1,2},y_{1,2})$ are
located on the hyperbola. We also observe that the singularities
$M_{3,4}(x_{3,4},y_{3,4})$ are located on the invariant line $y=0$.

On the other hand, for systems \eqref{sys:CS-1-IL} we calculate
\begin{gather*}
\chi_A^{(1)}=2^{-12} (h-1)^2 (2h-1)^2 (3h-1)^2 (1-2ah),\\
\chi_B^{(1)}=-105 a (h-1)^2 (2h-1)^2 (3h-1)^2/512,\\
\chi_C^{(1)}=2^{-16} 35ah (h-1)^4 (2h-1)^2 (2h+1)^2 (3h-1)^2, \\
{\rm D}=3a^2(2h-1)^4\big[2a(2h-1)-1\big] (1-2ah),
\end{gather*}
and it is clear that, because of the factors $1-2ah$ and $1+2a(1-2h)$,
the invariant polynomials $\chi_A^{(1)}$ and ${\rm D}$ govern the
types of the above singular points (i.e. are they real or complex
or coinciding), whereas the invariant polynomials $\chi_B^{(1)}$
and $\chi_C^{(1)}$ are respectively responsible for the position
of the hyperbola on the plane and for the location of the real
singularities on the hyperbola (i.e. on the same branch or on the
different ones).


($\alpha1$) {\it Subcase $\chi_A^{(1)}<0$.} Then the
singularities $M_{1,2}$ (located on the hyperbola) are complex,
whereas the types of singularities $M_{3,4}$ (located on the
invariant line $y=0$) are governed by ${\rm D}$.
We observe that clearly the condition $\chi_A^{(1)}<0$ implies
$\chi_C^{(1)}>0$.

Furthermore, we see that $\chi_B^{(1)}>0$ implies ${\rm D}<0$.
Indeed, the condition $\chi_B^{(1)}>0$ yields $a<0$ and, since $1-2ah<0$
(i.e. $4ah>2$), we have $2a(2h-1)-1=4ah-2a-1>0$; then ${\rm D}<0$.
So we arrive at the following conditions and configurations:
\begin{itemize}
 \item $\chi_B^{(1)}<0$ and ${\rm D}<0$ $\Rightarrow$ Config. H.39;
 \item $\chi_B^{(1)}<0$ and ${\rm D}>0$ $\Rightarrow$ Config. H.49;
 \item $\chi_B^{(1)}<0$ and ${\rm D}=0$ $\Rightarrow$ Config. H.44;
 \item $\chi_B^{(1)}>0$ $\Rightarrow$  Config. H.38.
\end{itemize}


($\beta$) {\it Subcase $\chi_A^{(1)}>0$.} In this case the
singularities $M_{1,2}$ are real and we have to decide if they are
located either on different branches or on the same branch and,
 moreover, the position of the hyperbola.

We observe that the conditions $\chi_B^{(1)}<0$ and $\chi_C^{(1)}<0$
imply ${\rm D}<0$. Indeed, the conditions $\chi_B^{(1)}<0$ and $\chi_C^{(1)}<0$ yield $a>0$ and $ah<0$, respectively, and, since $1-2ah>0$, we have $2a(2h-1)-1=4ah-2a-1<0$; then ${\rm D}<0$.

So in the case $\chi_B^{(1)}<0$ we get the following conditions and configurations:
\begin{itemize}
 \item $\chi_C^{(1)}<0$  $\Rightarrow$ Config. H.75;
 \item $\chi_C^{(1)}>0$ and ${\rm D}<0$  $\Rightarrow$ Config. H.74;
 \item $\chi_C^{(1)}>0$ and ${\rm D}>0$  $\Rightarrow$ Config. H.48;
 \item $\chi_C^{(1)}>0$ and ${\rm D}=0$  $\Rightarrow$ Config. H.64;
\end{itemize}
whereas for $\chi_B^{(1)}>0$ we get
\begin{itemize}
 \item $\chi_C^{(1)}<0$ and ${\rm D}<0$  $\Rightarrow$ Config. H.73;
$\chi_C^{(1)}>0$ and ${\rm D}<0$  $\Rightarrow$ Config. H.72;

 \item $\chi_C^{(1)}<0$ and ${\rm D}>0$  $\Rightarrow$ Config. H.47;
 $\chi_C^{(1)}>0$ and ${\rm D}>0$  $\Rightarrow$ Config. H.46;

 \item $\chi_C^{(1)}<0$ and ${\rm D}=0$  $\Rightarrow$ Config. H.66;
$\chi_C^{(1)}>0$ and ${\rm D}=0$  $\Rightarrow$ Config. H.65.
\end{itemize}

($\gamma$) {\it Subcase $\chi_A^{(1)}=0$.} By condition \eqref{cond:CS-1},
the condition $\chi_A^{(1)}=0$ implies $Z_1=0$ and hence $a=1/(2h)$.
In this case the points $M_{1,2}$ coalesce and we have a double point on
the hyperbola. For systems \eqref{sys:CS-1} with $a=1/(2h)$ we calculate
\begin{gather*}
 \chi_C^{(1)}=\frac{35}{2^{17}}\,(h-1)^4 (2h-1)^2 (2h+1)^2 (3h-1)^2,\\
 {\rm T}=\frac{3}{2^{10}h} (h-1) (2h-1)^4 y^2 \big[x^2+4h(h-1)y^2\big]^2.
\end{gather*}

From \eqref{cond:CS-1-IL}, we have $\chi_C^{(1)}>0$ and
$\operatorname{sign}({\rm T})= \operatorname{sign}\big(h(h-1)\big)$, therefore according to
Proposition~\ref{prop:number-FSPs}, besides the double point on
the hyperbola, we could have two simple points on the invariant
line $y=0$.

We observe that the condition $\chi_B^{(1)}>0$ implies ${\rm T}>0$.
Indeed, if $\chi_B^{(1)}>0$ we have $a<0$ and, since $a=1/(2h)$ (i.e. $h<0$),
we obtain $h(h-1)>0$; then ${\rm S}>0$.

So we arrive at the configuration Config. H.67 if
$\chi_B^{(1)}<0$ and ${\rm T}<0$; Config. H.43 if
$\chi_B^{(1)}<0$ and ${\rm T}>0$; and Config. H.42 if
$\chi_B^{(1)}>0$.


(a2) {\it Case $\mu_0=0$.} Then $h(2h-1)=0$ and considering the
condition \eqref{cond:CS-1-IL} we get $h=0$. In this case one of the
singular point located on the hyperbola has gone to infinity and
coalesced with $[0:1:0]$ (since $\mu_1= x/4$, see Lemma \ref{lem:mu_i-ISPs}).
 The second singularity on the hyperbola has the coordinates $(-2,\,-a/2)$,
 whereas the coordinates of the singularities
$M_{3,4}(x_{3,4},y_{3,4})$ located on the invariant line $y=0$
remain the same. Since for systems \eqref{sys:CS-1-IL} with $h=0$
we have ${\rm D}=-3a^2 (2a+1)$ we obtain $\operatorname{sign}({\rm D})=\operatorname{sign}(2a+1)$.

We observe that in the case $\chi_B^{(1)}<0$, we have $a>0$ and
hence ${\rm D}=2a+1>0$, which implies the existence of two real simple
singularities on $y=0$ and we obtain the configuration shown in
Config. H.70. Now, in the case $\chi_B^{(1)}>0$, we
obtain the following conditions and configurations:
Config. H.71 if ${\rm D}<0$; Config. H.41 if
${\rm D}>0$; and Config. H.55 if ${\rm D}=0$.


(b) {\it Possibility $\chi_E^{(1)}=0$.} In this case we
obtain $a=1/(g+h)^2$ and this leads to the systems
\begin{equation} \label{sys:CS-1-ILa}
\begin{gathered}
 \frac {dx}{dt}= \frac{2h-1}{(g+h)^2}+ x+gx^2+(h-1)xy,\\
 \frac {dy}{dt}= \frac{2g-1}{(g+h)^2}- y+(g-1)xy+hy^2
\end{gathered}
\end{equation}
possessing the following invariant line and invariant hyperbola
\begin{equation} \label{hyp:H1+IL}
 x-y +2/(g+h)=0,\quad \Phi(x,y)=\frac{1}{(g+h)^2}+xy=0.
\end{equation}

We claim that the condition $\chi_E^{(1)}=0$ implies ${\rm D}\leq0$ and
 $\chi_B^{(1)}<0$. Indeed, if $\chi_E^{(1)}=0$, then
 $a=1/(g+h)^2$ and in this case we see that
\begin{gather*}
\chi_B^{(1)}=-\frac{105(g-1)^2(h-1)^2(g-h)^2(3g-1)^2(3h-1)^2}{8(g+h)^2}<0,\\
{\rm D}=-\frac{192(g-h)^6 (g+h-2)^2 (g+h-2gh)^2}{(g+h)^8}\leq0,
\end{gather*}
by condition \eqref{cond:CS-1-IL}. This proves our claim.

For the above systems we calculate
\begin{equation} \label{val:B2-CS1-ILa}
B_2=-\frac{648}{(g+h)^4}\,(g-1)^2 (h-1)^2 (2g-1) (2h-1) (x-y)^4
\end{equation}
and by Lemma \ref{lem:Bi-0} for the existence of an invariant line
in a direction different from $y=x$ it is necessary $B_2=0$.


(b1) {\it Case $B_2\ne0$.} Since $\theta\ne0$ by Lemma \ref{lem:theta,N}
 we could not have a couple of parallel invariant lines in the direction
$y=x$ and obviously the invariant line $y=x+2/(g+h)$ is a simple one.
As before we consider two subcases: $\mu_0\ne0$ and $\mu_0=0$.


($\alpha1$) {\it Subcase $\mu_0\ne0$.}
Then $gh(g+h-1)\ne0$ and systems \eqref{sys:CS-1-ILa} possess four real
singularities $M_i(x_i,y_i)$ with the coordinates
\begin{equation} \label{coord:FSPs-CS-1-ILa}
\begin{gathered}
x_1=-\frac{1}{g+h},\quad y_1=\frac{1}{g+h};\quad
x_2=-\frac{h}{g(g+h)},\quad y_2= \frac{g}{h(g+h)};\\
x_3=-\frac{2h-1}{g+h},\quad y_3=\frac{2g-1}{g+h};\\
x_4=-\frac{2h-1}{(g+h)(g+h-1)},\quad y_4=\frac{2g-1}{(g+h)(g+h-1)}.
\end{gathered}
\end{equation}
It could be checked directly that the singularity $M_1(x_1,y_1)$
is a common (tangency) point of the invariant hyperbola and of
the invariant line \eqref{hyp:H1+IL}. Moreover, the singular point
$M_2(x_2,y_2)$ (respectively $M_4(x_4,y_4)$) is located on the
hyperbola (respectively on the invariant line), whereas the
singularity $M_3(x_3,y_3)$ generically is located outside the
invariant hyperbola as well as outside the invariant line.

For systems \eqref{sys:CS-1-ILa} we calculate
\begin{equation} \label{val:D,chi-j,CS-1-ILa}
\begin{gathered}
\chi_A^{(1)}=-\frac{1}{64} (g-1)^2(h-1)^2 (g-h)^2 (3g-1)^2
(3h-1)^2\,Z_1\big|_{\{a=1/(g+h)^2\}},\\
\chi_C^{(1)}=\frac{35}{32} gh(g-1)^4(h-1)^4 (g-h)^2 (3g-1)^2 (3h-1)^2,\\
\chi_D^{(1)}=-\frac{105}{2(g+h)^2} (g-h)^3 (3g-1)(3h-1) (g+h-2gh)
\end{gathered}
\end{equation}
and, by \eqref{cond:CS-1}, the condition $\chi_A^{(1)}=0$ is
equivalent to $Z_1=-(g-h)^2/(g+h)^2=0$ and this contradicts the
condition \eqref{cond:CS-1}. So the singular points $M_1$ and
$M_2$ could not coalesce.

We consider two possibilities: $\chi_C^{(1)}<0$ and $\chi_C^{(1)}>0$.


($\alpha1$) {\it Possibility $\chi_C^{(1)}<0$.} In
this case the singularities $M_{1,2}$ are located on different
branches of the hyperbola and we need to decide if the singular
point $M_3$ coalesces with the singularities on the hyperbola, and
this fact is governed by the polynomial ${\rm D}$. However, this last
polynomial could vanish because of the factors $g+h-2$ and $g+h-2gh$.
Then, according to \eqref{val:D,chi-j,CS-1-ILa}, we need to
distinguish the cases $\chi_D^{(1)}\ne0$ and $\chi_D^{(1)}=0$.

So we get the configurations Config. H.60 if ${\rm D}\ne0$;
Config. H.63 if ${\rm D}=0$  and $\chi_D^{(1)}\ne0$; and
Config. H.69 if ${\rm D}=0$ and $\chi_D^{(1)}=0$.


($\alpha2$) {\it Possibility $\chi_C^{(1)}>0$.} Assume $\chi_C^{(1)}>0$,
 i.e. $gh>0$. Then, by Lemma~\ref{lem:sign(x1*x2)}, both singularities 
$M_{1,2}$ are located on the same branch of hyperbola. 
It is clear that the reciprocal position of the singularities $M_2$ 
(located on the hyperbola) and $M_4$ (located on the invariant line) 
with respect to the tangency point $M_1$ of the hyperbola and the invariant 
line \eqref{hyp:H1+IL}, define different configurations. More exactly 
the type of the configuration depends on the sign of the expression:
$$
 (x_1-x_2)(x_1-x_4)=\frac{(g-h)^2}{g(g+h-1)(g+h)^2}
$$
and hence we need $\operatorname{sign}\big(g(g+h-1)\big)$ when $gh>0$.
 We calculate
$$
\chi_F^{(1)}=(g+h)(g+h-1)(g-1)^4(h-1)^4 (g-h)^2 (3g-1)^2(3h-1)^2/256
$$
and, since in the case $gh>0$ we have 
$\operatorname{sign}(g)=\operatorname{sign}(g+h)$, we deduce that
$$
\operatorname{sign}(\chi_F^{(1)})=\operatorname{sign}\big((g+h)(g+h-1)\big)
=\operatorname{sign}\big(g(g+h-1)\big).
$$

We observe that conditions $\chi_C^{(1)}>0$ and
$\chi_F^{(1)}<0$ imply ${\rm D}\ne0$ (i.e. ${\rm D}<0$). Indeed, if we
suppose ${\rm D}=0$, then $(g+h-2)(g+h-2gh)=0$. In the case $g=2-h$,
we have
$$
\chi_F^{(1)}=(h-1)^{10}(3h-5)^2(3h-1)^2/32>0,
$$
by \eqref{cond:CS-1-IL}, which contradicts the condition $\chi_F^{(1)}<0$.
On the other hand, if $g=h/(2h-1)$, we have
$$
\chi_F^{(1)}=\frac{1}{32(2h-1)^{10}}\,h^4 (h-1)^{10}(h+1)^2 (3h-1)^2 (1-2h+2h^2)>0,
$$
by \eqref{cond:CS-1-IL}, which again contradicts the condition
$\chi_F^{(1)}<0$. So we detect that in the case $\chi_F^{(1)}<0$
we obtain the configuration Config. H.61.

In the case $\chi_F^{(1)}>0$, the polynomial ${\rm D}$ could vanish
and we need to detect to which singular points $M_2$ or $M_4$ the
singularity $M_3$ collapses. So we get the following conditions
and configurations: Config. H.59 if ${\rm D}\ne0$;
Config. H.62 if ${\rm D}=0$ and $\chi_D^{(1)}\ne0$; and
Config. H.68 if ${\rm D}=0$ and $\chi_D^{(1)}=0$.


($\beta$) {\it Subcase $\mu_0=0$.} Then $gh(g+h-1)=0$ and, by
Lemma \ref{lem:mu_i-ISPs}, at least one finite singularity has gone to
infinity and coalesced with an infinite singular point.
Since for systems \eqref{sys:CS-1-ILa} we have $\chi_C^{(1)}=0$
if and only if $gh=0$ (see \eqref{val:D,chi-j,CS-1-ILa}), we consider
two possibilities: $\chi_C^{(1)}\ne0$ and $\chi_C^{(1)}=0$.


($\beta_1$) {\it Possibility $\chi_C^{(1)}\ne0$.} Then the condition $\mu_0=0$
implies $g+h-1=0$, i.e. $g=1-h$ and considering the coordinates
\eqref{coord:FSPs-CS-1-ILa} of the finite singularities of systems
\eqref{sys:CS-1-ILa} we observe that the singular point $M_4$
located on the invariant line has gone to infinity and coalesced
with the singularity $[1:1:0]$. In this case calculation yields
\begin{gather*}
\chi_A^{(1)}=h^2 (h-1)^2 (2h-1)^4 (3h-1)^2 (3h-2)^2/64,\\
\chi_B^{(1)}=-105\,h^2 (h-1)^2 (2h-1)^2 (3h-1)^2 (3h-2)^2/8,\\
\chi_C^{(1)}=35\,h^5 (1-h)^5 (2h-1)^2 (3h-1)^2 (3h-2)^2/32,\\
{\rm D}=-192 (2h-1)^6 (1-2h+2h^2)^2,
\end{gather*}
and by \eqref{cond:CS-1-IL} we have $\chi_A^{(1)}>0$,
$\chi_B^{(1)}>0$ and ${\rm D}<0$. Moreover, since by Remark
\ref{rem:mu0=0,mu1-ne0} the condition ${\rm R}\ne0$ holds, then
according to Proposition \ref{prop:number-FSPs} all three finite
singularities are distinct. This means that the singularities
located on the hyperbola are simple and belong to different
branches (respectively of the same branch) of the hyperbola if
$\chi_C^{(1)}<0$ (respectively $\chi_C^{(1)}>0$). As a result we
get configurations Config. H.56 if $\chi_C^{(1)}<0$ and
Config. H.57 if $\chi_C^{(1)}>0$.


($\beta2$) {\it Possibility $\chi_C^{(1)}=0$.} Then
$gh=0$ (this implies $\mu_0=0$) and we have $g^2+h^2\ne0$ because
$g-h\ne0$. Considering Remark \ref{rem:g<->h}, without loss of
generality, we may assume $g=0$. In this case, the singularity
$M_2$ located on the hyperbola \eqref{hyp:H1+IL} has gone to
infinity and coalesced with the point $[1,0,0]$. Since by Remark
\ref{rem:mu0=0,mu1-ne0} we have $\mu_1\ne0$, then according to
Lemma \ref{lem:mu_i-ISPs} other three finite singular points
remain on the finite part of the phase plane.

It is clear that depending on the position of the singular point
$M_4$ (located on the invariant line \eqref{hyp:H1+IL}) with
respect to the vertical line $x=x_1$ we get different
configurations. So this distinction is governed by the sign of the
expression $x_4-x_1=1/(1-h)$. Moreover, since in this case we have
the invariant line $x-y+2/h=0$, its position depends on the sign
of $h$. Then we need to control the $\operatorname{sign}\big(h(1-h)\big)$. Thus,
we calculate
$$
\chi_F^{(1)}=h^3(h-1)^5(3h-1)^2/256,\quad {\rm D}=-192 (h-2)^2
$$
and we have $\operatorname{sign}\big(h(1-h)\big)
=-\operatorname{sign}(\chi_F^{(1)})$.

It is clear that, in the case $\chi_F^{(1)}<0$, we have ${\rm D}\ne0$
and, since the condition ${\rm R}\ne0$ holds (see Remark
\ref{rem:mu0=0,mu1-ne0}), Proposition \ref{prop:number-FSPs}
assures us that all three finite singularities are distinct if
${\rm D}\ne0$. So we arrive at the configuration given by
Config. H.50.

Now, in the case $\chi_F^{(1)}>0$, the polynomial ${\rm D}$ could
vanish and we obtain the configuration Config. H.51 if
${\rm D}\ne0$ and Config. H.54 if ${\rm D}=0$.


(b2) {\it Case $B_2=0$.} Considering \eqref{val:B2-CS1-ILa}
 and the condition \eqref{cond:CS-1} we obtain $g=1/2$ and this
leads to the 1-parameter family of systems
\begin{equation} \label{sys:CS-1-ILb}
\frac {dx}{dt}= \frac{4(2h-1)}{(2h+1)^2}+ x+\frac{x^2}{2}+(h-1)xy,\quad
 \frac {dy}{dt}= -y (2+x-2hy)/2,
\end{equation}
for which the condition $\theta\beta_1\beta_2\ne0$ gives
\begin{equation} \label{cond:CS-1-ILb}
(h-1)(2h+1)(2h-1)(3h-1)(6h-5)\ne0.
\end{equation}
These systems possess two invariant lines and a hyperbola
$$
x-y+\frac{4}{2h+1}=0,\quad y=0,\quad \Phi(x,y)=\frac{4}{(2h+1)^2}+xy=0.
$$
as well as the following singularities $M_i(x_i,y_i)$:
\begin{equation} \label{coord:FSPs-CS-1-ILb}
\begin{gathered}
x_1=-\frac{2}{2h+1},\quad  y_1= \frac{2}{2h+1};\quad
x_2=-\frac{4h}{2h+1},\quad y_2= \frac{1}{h(2h+1)};\\
x_3= \frac{2(1-2h)}{2h+1},\quad y_3= 0;\quad
x_4=-\frac{4}{2h+1},\quad y_4=0.
\end{gathered}
\end{equation}
We observe that  from condition \eqref{cond:CS-1-ILb} all
singularities are located on the finite part of the phase plane,
except the singular point $M_2$ which could go to infinity in the
case $h=0$. For the above systems we calculate
$$
\chi_C^{(1)}=35h (h-1)^4 (2h-1)^2 (3h-1)^2/16384
$$
and we analyze the subcases $\chi_C^{(1)}<0$, $\chi_C^{(1)}>0$ and 
$\chi_C^{(1)}=0$.


($\alpha1$) {\it Subcase $\chi_C^{(1)}<0$.} Then $h<0$ and it implies
$$
\mu_0=h(2h-1)/4\ne0, \quad {\rm D}=-\frac{48}{(2h+1)^8}\,(2h-3)^2(2h-1)^6\ne0.
$$
Since the singular points on the hyperbola are located on
different branches, we arrive at the unique configuration
Config. H.84.

($\beta$) {\it Subcase $\chi_C^{(1)}>0$.} Then $h>0$
(this implies again $\mu_0\ne0$) and the singularities on the
hyperbola are located on the same branch. Thus, it is necessary to
distinguish the position of $M_2$ on the hyperbola with relation
to $M_1$, which is the intersection point of the hyperbola and the
invariant line $x-y+4/(2h+1)=0$, and $M_4$, which is the
intersection point of the two invariant lines, as well as the
position of the singularities $M_3$ and $M_4$ on the invariant
line $y=0$. We calculate
$$
(x_1-x_2)(x_1-x_4)=\frac{4(2h-1)}{(2h+1)^2},\quad
(x_4-x_3)=\frac{2(2h-3)}{2h+1}
$$
and hence $\operatorname{sign}(2h-1)$ 
(respectively $\operatorname{sign}(2h-3)$) will describe
the position of the singularity $M_2$ on the hyperbola (respectively
the position of the singularity $M_3$ on the invariant line $y=0$).
We calculate
$$
\chi_F^{(1)}=2^{-18}\,(2h-1)^3(2h+1)(h+1)^4(3h-1)^2,\quad
\chi_G^{(1)}=(2h-3)(h+1)/8
$$
and, by \eqref{cond:CS-1-ILb} and since $h>0$, we obtain
$\operatorname{sign}(2h-1)=\operatorname{sign}(\chi_F^{(1)})$ and 
$\operatorname{sign}(2h-3)=\operatorname{sign}(\chi_G^{(1)})$.

We observe that the condition $\chi_G^{(1)}=0$ yields $h=3/2$ and this
implies ${\rm D}=0$. In this sense, we obtain the following
conditions and configurations:
\begin{itemize}
 \item $\chi_F^{(1)}<0$  $\Rightarrow$  Config. H.86;
 \item $\chi_F^{(1)}>0$ and $\chi_G^{(1)}<0$  $\Rightarrow$ Config. H.85;
 \item $\chi_F^{(1)}>0$ and $\chi_G^{(1)}>0$  $\Rightarrow$ Config. H.83;
 \item $\chi_F^{(1)}>0$ and $\chi_G^{(1)}=0$  $\Rightarrow$ Config. H.81;
\end{itemize}


($\gamma$) {\it Subcase $\chi_C^{(1)}=0$.} Then
$h=0$ (this implies $\mu_0=0$) and the singularity $M_2$ has gone
to infinity and coalesced with $[0:1:0]$. As a result we get
Config. H.82.

\paragraph{Case $\beta_2=0$.} Since $\beta_1\ne0$ (i.e. $c\ne0$) we get
$(g-h)(3g+3h-4)=0$. On the other hand, for systems
\eqref{sys:eta-poz-theta-ne0-gamma1=0} we have
$$
\beta_3=-c(g-h)(g-1)(h-1)/4
$$
and we consider two possibilities: $\beta_3\ne0$ and $\beta_3=0$.

\subparagraph{Possibility $\beta_3\ne0$.} In this case we
have $g-h\ne0$ and the condition $\beta_2=0$ yields $3g+3h-4=0$,
i.e. $g=4/3-h$. In this case, for systems
\eqref{sys:eta-poz-theta-ne0-gamma1=0}, we calculate
\begin{gather*}
\gamma_3=7657c (h-1)^3 (3h-1)^3 \big[a(5-6h)-3b(2h-1)\big],\\
\beta_3=-c (h-1) (3h-2) (3h-1)/18,\quad
\mathcal{R}_1=(a-b) c (h-1)^3 (3h-1)^3/6.
\end{gather*}
Without loss of generality, we may assume $2h-1\ne0$, otherwise
via the change $(x,y,t,a,b,c)\mapsto(y,x,t,b,a,-c)$ we could
bring systems \eqref{sys:eta-poz-theta-ne0-gamma1=0} with $h=1/2$
to the same systems with $h=5/6$. Therefore, from $\beta_3\ne0$,
the condition $\gamma_3=0$ yields $b=a(5-6h)/[3(2h-1)]$ and since
$c\ne0$ we may assume $c=1$ by the rescaling
$(x,y,t)\mapsto (cx,cy,t/c)$.

We remark that the condition $\gamma_3=0$ could be written as $a=a_1(2h-1)$
and $b=a_1(5-6h)/3$. So setting the old parameter $a$ instead of $a_1$,
we arrive at the 2-parameter family of systems
\begin{equation} \label{sys:CS-2}
\begin{gathered}
 \frac {dx}{dt}= 3a(2h-1)+ x+\frac{4-3h}{3}\,x^2+(h-1)xy,\\
 \frac {dy}{dt}= \frac{a(5-6h)}{3}- y+ \frac{1-3h}{3}\,xy+hy^2,
\end{gathered}
\end{equation}
for which the condition $\theta\beta_1\beta_3\mathcal{R}_1\ne0$ is
equivalent to the condition
\begin{equation} \label{cond:CS-2}
a (h-1) (3h-1) (3h-2) \ne0.
\end{equation}
Moreover, these systems possess the same invariant hyperbola \eqref{Hyp:1}.

\begin{observation} \label{obs3} \rm
We observe that the family of systems \eqref{sys:CS-2} is in fact
a subfamily of systems \eqref{sys:CS-1} under the relation
$g=4/3-h$. Moreover, if we present the condition \eqref{cond:CS-1}
in the form $F(a,g,h)(3g+3h-4)\ne0$, then in the case $g=4/3-h$,
the condition \eqref{cond:CS-2} is equivalent to $F(a,g,h)\ne0$.
We also point out that the condition $g=4/3-h$ does not imply the
vanishing of any of the invariants
$\chi_A^{(1)},\chi_B^{(1)},\ldots,\chi_G^{(1)}$. Hence, all the
configurations of systems \eqref{sys:CS-2} are the configurations
of systems \eqref{sys:CS-1} determined by the same invariant
conditions.
\end{observation}

Considering this observation, we could join the conditions
defining the family \eqref{sys:CS-1} (i.e. $\eta>0$,
$\theta\beta_1\beta_2\ne0$) with the conditions which define the
subfamily \eqref{sys:CS-2} (i.e. $\eta>0$, $\theta\beta_1\ne0$,
$\beta_2=0$ and $\beta_3\ne0$). More precisely, the conditions
defining both such families of systems are
$\beta_2^2+\beta_3^2\ne0$ and $(\mathfrak{C}_1)$, where
$$
 (\mathfrak{C}_1): (\beta_2\mathcal{R}_1\ne0) \cup 
(\beta_2=\gamma_3=0\cap \beta_3\ne0).
$$

\subparagraph{Possibility $\beta_3=0$.} From $\beta_1\ne0$
(i.e. $(g-1)(h-1)\ne0$), we get $g=h$. In this case, we calculate
\begin{gather*}
\gamma_2= 6300 h (h-1)^4 (3h-2) (3h-1)^2 \mathcal{B}_1,\\
\theta= -h(h-1)^2,\quad \beta_1= -(h-1)^2 (3h-1)^2/4,\\
\beta_4= 2 h (3h-2),\quad \beta_5= -2 h^2 (2h-1).
\end{gather*}
We shall consider two cases: $\beta_4\ne0$ and $\beta_4=0$.


(a) {\it Case $\beta_4\ne0$.} So the condition $\gamma_2=0$ implies
 $\mathcal{B}_1=0$ and by Theorem~\ref{trm:exist-hyp} the condition
 $\beta_5\ne0$ is necessary for the existence of hyperbola. Hence,
we arrive at the particular case of systems~\eqref{sys:CS-1} when $g=h$,
i.e. we get the systems
\begin{equation} \label{sys:CS-3}
\begin{gathered}
 \frac {dx}{dt}= a(2h-1) + x + hx^2 + (h-1)xy,\\
 \frac {dy}{dt}= a(2h-1) - y + (h-1)xy + hy^2
\end{gathered}
\end{equation}
with the condition
\begin{equation} \label{cond:CS-3}
a h (h-1) (2h-1) (3h-1) (3h-2) \ne0.
\end{equation}
These systems possess the invariant line and hyperbola
$$
1+h(x-y)=0,\quad \Phi(x,y)=a+xy=0.
$$
Since $\mu_0=h^2 (2h-1)\ne0$ (see \eqref{cond:CS-3}), systems have finite
singularities $M_{i}\big(x_{i},y_{i}\big)$ of total multiplicity 4:
%\label{page:x_i,y_i-a}
\begin{gather*}
x_{1,2}=-\frac{1\pm\sqrt{1-4ah^2}}{2h},\quad
y_{1,2}=\frac{1\mp\sqrt{1-4ah^2}}{2h},\\
x_{3,4}=\frac{-1\pm\sqrt{1+4a-8ah}}{2},\quad
y_{3,4}=\frac{1\mp\sqrt{1+4a-8ah}}{2},
\end{gather*}

We detect that the singularities $M_{1,2}$ are located on both the hyperbola
 and the straight line. These singular points are located on different
branches (respectively on the same branch) of the hyperbola if only
if $x_1x_2<0$ (respectively $x_1x_2>0$), where $x_1x_2=a$.
Moreover, these singularities are real if $1-4ah^2>0$, they are complex
if $1-4ah^2<0$ and they coincide if $1-4ah^2=0$.

On the other hand, we calculate
$$
\chi_A^{(2)}=2h^2 (2h-1)^2 (3h-1)^2 (1-4ah^2),\quad
\chi_B^{(2)}=-a (h-1)^2 (2h-1)^2 (3h-1)^4/4
$$
and, by condition \eqref{cond:CS-3}, we have
$\operatorname{sign}(1-4ah^2)=\operatorname{sign}(\chi_A^{(2)})$ 
(if $1-4ah^2\ne0$) and
$\operatorname{sign}(x_1x_2)=-\operatorname{sign}(\chi_B^{(2)})$.

We observe that at least one of the singular points $M_{3,4}$ could be
located either on the invariant hyperbola or on the invariant straight line.
 Next we determine the conditions for this to happen. We calculate
\begin{gather*}
 \Phi(x,y)|_{\{x=x_{3,4},\, y=y_{3,4}\}}=(-1+4ah\pm\sqrt{1+4a-8ah})
\equiv \Omega'_{3,4}(a,h),\\
 [1+h(x-y)]|_{\{x=x_{3,4},\, y=y_{3,4}\}}=1+h(-1\pm\sqrt{1+4a-8ah})
\equiv  \Theta_{3,4}(a,h).
\end{gather*}
So $M_3$ or $M_4$ could be located on the invariant hyperbola
(respectively invariant line) if and only if $\Omega'_3 \Omega'_4=0$
(respectively $\Theta_3 \Theta_4=0$). So we have
$$
\Omega'_3 \Omega'_4=-a(1-4ah^2)=0,\quad
\Theta_3 \Theta_4=(1-2h)(1-4ah^2)=0
$$
if and only $1-4ah^2=0$ (because of condition \eqref{cond:CS-3}).

Thus, in the case $\chi_A^{(2)}\ne0$ we arrive at the
configuration given by Config. H.37 if $\chi_A^{(2)}<0$;
Config. H.52 if $\chi_A^{(2)}>0$ and $\chi_B^{(2)}<0$;
and Config. H.53 if $\chi_A^{(2)}>0$ and
$\chi_B^{(2)}>0$.

Assume now $\chi_A^{(2)}=0$, i.e. $1-4ah^2=0$.
By condition \eqref{cond:CS-3} we have $h\ne0$ and hence $a=1/(4h^2)$.
 It could be easily observed that in this case the singular points
$M_2$ and $M_3$ coalesce with the singularity $M_1$ and this point
becomes a triple point of contact of the invariant hyperbola and invariant line.
 We remark that this point of contact could not be of multiplicity
4 because in this case we have
$$
\mu_0=h^2 (2h-1)\ne0,\quad {\rm D}={\rm T}={\rm P}=0,\quad
{\rm R}=3h^2 (h-1)^2 (2h-1)^2 (x+y)^2\ne0,
$$
by condition \eqref{cond:CS-3}. Thus, in the case
$\chi_A^{(2)}=0$ we get the configuration given by
Config. H.45.


(b) {\it Case $\beta_4=0$.} Then, from $\theta\ne0$, we get $h=2/3$
and we obtain a family of systems which is a subfamily of systems
\eqref{sys:CS-3} setting $h=2/3$. Since in this case we have
$$
\chi_A^{(2)}=8(9-16a)/729,\quad \chi_B^{(2)}=-a/324,
$$
it is clear that we obtain again the same four configurations as for
the family \eqref{cond:CS-3} with the same invariant conditions.
As earlier we could join the cases $\beta_4\ne0$ and $\beta_4=0$.
 More precisely, the conditions defining the corresponding families
of systems are
$$
 (\mathfrak{C}2): \ (\beta_4\beta_5\mathcal{R}_2\ne0) \cup (\beta_4=\gamma_3=0,
\; \mathcal{R}_2\ne0).
$$

\subsubsection {The possibility $\beta_1=0$}\label{subsec:CS-4}

Considering \eqref{val:gamma1=0} and the condition $\theta\ne0$, we get
$c(3g-1)(3h-1)=0$. On the other hand, for systems
\eqref{sys:eta-poz-theta-ne0-gamma1=0} we calculate
$$
\beta_6=-c (g-1) (h-1)/2
$$
and we shall consider two cases: $\beta_6\ne0$ and $\beta_6=0$.

\paragraph{Case $\beta_6\ne0$.}
Then $c\ne0$ (as before we could assume $c=1$ by a rescaling) and the
 condition $\beta_1=0$ implies $(3g-1)(3h-1)=0$. Therefore, by
 Remark \ref{rem:g<->h}, we may assume $h=1/3$ and this leads to the
 following 3-parameter family of systems
\begin{equation} \label{sys:eta-poz-theta-ne0-gamma1=0c}
\frac {dx}{dt}= a+x+ gx^2 -2xy/3,\quad
 \frac {dy}{dt}=b-y+ (g-1)xy+ y^2/3,
\end{equation}
which is a subfamily of \eqref{sys:eta-poz-theta-ne0-gamma1=0}.

For these systems we calculate
\begin{gather*}
\begin{aligned}
\gamma_4
&=16(g-1)^2(3g-1)^2\big[3a(2g-1)+b\big]\big[(3g+1)^2 (b-2a+6ag) \\
&\quad + 6(1-3g)\big]/243,
\end{aligned}\\
\beta_6=(g-1)/3,\quad \beta_2=(1-g) (3g-1)/2,\quad
\mathcal{R}_3=a (3g-1)^3/18.
\end{gather*}

\subparagraph{Subcase $\beta_2\ne0$.} Then $3g-1\ne0$ and,  to have $\gamma_4=0$,
we must have $\big[3a(2g-1)+b\big]\big[(3g+1)^2 (b-2a+6ag) + 6(1-3g)\big]=0$.

We claim that systems \eqref{sys:eta-poz-theta-ne0-gamma1=0c} with
$(3g+1)^2 (b-2a+6ag) + 6(1-3g)=0$ (i.e. $b=2(3g-1) (3-a-6ag-9ag^2)/(3g+1)^2$)
 could be brought to the same systems with $b=3a(1-2g)$ via an affine
 transformation. Indeed, from $\theta\ne0$ (i.e. $(3g+1)(g-1)\ne0$),
 we may apply the affine transformation
\begin{equation} \label{transf:gamma4=0}
x_1=\frac{3g+1}{3(1-g)}x,\quad y_1=\frac{3g+1}{3(1-g)}(x-y)+\frac{2}{1-g},\quad
 t_1=\frac{3(g-1)}{3g+1}t,
\end{equation}
and we arrive at the systems
$$
\frac {dx_1}{dt_1}= a_1 + x_1 + g_1 x_1^2 - 2 x_1y_1/3,\quad
 \frac {dy_1}{dt_1}=b_1 - y_1 + (g_1-1) x_1 y_1 + y_1^2/3,
$$
where $b_1=-3 a_1 (2g_1-1)$, $a_1=-a (3g+1)^2/[9(g-1)^2]$ and
$g_1=(2-3g)/3$. This completes the proof of our claim.

Next we consider the following family of systems
 \begin{equation} \label{sys:CS-4}
\begin{gathered}
 \frac {dx}{dt}= a+x+ gx^2 -2xy/3,\\
 \frac {dy}{dt}=-3 a (2g-1)-y+ (g-1)xy+ y^2/3,
\end{gathered}
\end{equation}
with the condition
\begin{equation} \label{cond:CS-4}
a (g-1) (3g-1) (3g+1) \ne0.
\end{equation}

According to Theorem \ref{trm:exist-hyp}, these systems possess either
one or two invariant hyperbolas if either $\delta_1\ne0$ or $\delta_1=0$,
respectively, where $\delta_1=(3g-1)\big[6(1-3g)+a(3g+1)^2\big]/18$.


(a) {\it Possibility $\delta_1\ne0$.}
Then systems \eqref{sys:CS-4} possess the unique invariant hyperbola
\begin{equation} \label{Hyp:4}
\Phi(x,y)=3a-xy=0.
\end{equation}

For systems \eqref{sys:CS-4} we calculate
\begin{equation} \label{val:B1-CS4}
B_1= 8a^2 (g-1)^2 (2g-1) (3g-1) \big[3+a(3g+1)^2\big]/27.
\end{equation}


(a1) {\it Case $B_1\ne0$.} In this case, by
\eqref{cond:CS-4}, we have $(2g-1)\big[3+a(3g+1)^2\big]\ne0$.
For systems \eqref{sys:CS-4} we calculate $\mu_0=g(3g-2)/9$
and we consider two possibilities: $\mu_0\ne0$ and $\mu_0=0$.


($\alpha1$) {\it Subcase $\mu_0\ne0$.} In this case
the systems have finite singularities of total multiplicity 4
with coordinates $M_i(x_i,y_i)$:
 %\label{page:x_i,y_i-a}
\begin{gather*}
x_{1,2}=\frac{-1 \pm \sqrt{1+4ag}}{2g},\quad
y_{1,2}=\frac{3(1 \pm \sqrt{1+4ag})}{2},\\
x_{3,4}=\frac{1 \pm \sqrt{1-8a+12ag}}{2(3g-2)},\quad
y_{3,4}= \frac{3(2g-1)(1 \pm \sqrt{1-8a+12ag})}{2(3g-2)}.
\end{gather*}

We detect that the singularities $M_{1,2}$ are located on the
invariant hyperbola. More exactly, these singular points are
located on different branches (respectively on the same branch) of
the hyperbola if only if $x_1x_2<0$ (respectively $x_1x_2>0$),
where $x_1x_2=-a/g$. Moreover, these singularities are real if
$1+4ag>0$, complex if $1+4ag<0$ or they coincide if $1+4ag=0$.

On the other hand, we calculate
\begin{gather*}
\chi_A^{(3)}=\frac{7713280}{243} (1+4ag)\big[6(1-3g)+a(3g+1)^2\big]^2,\\
\chi_B^{(3)}=\frac{164798932}{81} a(g-1)^2(3g-1)^2\big[6(1-3g)+a(3g+1)^2\big]^2,\\
\chi_C^{(3)}=-\frac{66560}{9} ag\big[6(1-3g)+a(3g+1)^2\big]^2,
\end{gather*}
and, from condition \eqref{cond:CS-4}, we have
$\operatorname{sign}(\chi_A^{(3)})=\operatorname{sign}(1+4ag)$ (if $1+4ag\ne0$) and
$\operatorname{sign}(\chi_C^{(3)})=\operatorname{sign}(x_1x_2)$.

We point out that at least one of the singular points $M_{3,4}$
could be located on the invariant hyperbola. Next we determine
the conditions for this to happen. We calculate
\begin{align*}
\Phi(x,y)|_{\{x=x_{3,4},\, y=y_{3,4}\}}
&=\frac{3\big[2a(g-1)
(3g-2)-(2g-1) (1\pm\sqrt{1-8a+12ag})\big]}{2(3g-2)^2} \\
&\equiv \Omega''_{3,4}(a, g,h).
\end{align*}
It is clear that at least one of the singular points $M_3$ or
$M_4$ belongs to the hyperbola \eqref{Hyp:4} if and only if
$\Omega''_3 \Omega''_4=0$. So we have
$$
\Omega''_3 \Omega''_4=\frac{9a \big[a(g-1)^2-2g+1\big]}{(3g-2)^2}=0
$$
and, since
$$
\chi_D^{(3)}=\frac{736}{81}\,(3g-1)\big[a(g-1)^2-2g+1\big]\big[6(1-3g)
+ a(3g+1)^2\big],
$$
we deduce that at least one of the singular points $M_{3,4}$
belongs to the hyperbola if and only if $\chi_D^{(3)}=0$.

($\alpha1$) {\it Possibility $\chi_A^{(3)}<0$.} So
we have no real singularities located on the invariant hyperbolas
and we arrive at the configurations given by Config. H.1
if $\chi_B^{(3)}<0$ and Config. H.2 if $\chi_B^{(3)}>0$.


($\alpha2$) {\it Possibility $\chi_A^{(3)}>0$.} In
this case we have two real singularities located on the hyperbola
and we need to decide if they are located either on different
branches or on the same branch of the invariant hyperbola and also
if at least one of the singular points $M_{3,4}$ will belong to
the hyperbola.


(i) {\it Case $\chi_D^{(3)}\ne0$.} Then
$a(g-1)^2-2g+1\ne0$ and on the hyperbola there are two simple real
singularities (namely $M_{1,2}$) and we arrive at the conditions
and configurations given by:
\begin{itemize}
 \item $\chi_C^{(3)}<0$ and $\chi_B^{(3)}<0$  $\Rightarrow$ Config. H.17;
 \item $\chi_C^{(3)}<0$ and $\chi_B^{(3)}>0$  $\Rightarrow$ Config. H.19;
 \item $\chi_C^{(3)}>0$ and $\chi_B^{(3)}<0$  $\Rightarrow$ Config. H.16;
 \item $\chi_C^{(3)}>0$ and $\chi_B^{(3)}>0$  $\Rightarrow$ Config. H.18.
\end{itemize}


(ii) {\it Case $\chi_D^{(3)}=0$.} In this case,
because of $B_1\ne0$ and \eqref{cond:CS-4}, we obtain
$a=(2g-1)/(g-1)^2$. Then, considering Proposition
\ref{prop:number-FSPs}, we calculate
\begin{gather*}
{\rm D}=0, \\
\begin{aligned}
 {\rm T}&=-\frac{(5g-3)^2}{2187(g-1)^4}(3g-1)^2
(3gx-y)^2 \big[3(2g-1)x-y\big]^2 \\
&\quad\times \big[3g(g-1)x + 2(2g-1)y\big]^2.
\end{aligned}
\end{gather*}


(ii.1) {\it Subcase ${\rm T}\ne0$.} Then ${\rm T}<0$,
$\chi_A^{(3)}>0$ and, according to Proposition
\ref{prop:number-FSPs}, in this case systems \eqref{sys:CS-1}
possess one double and two simple real finite singularities. More
exactly, we detect that one of the singular points $M_3$ or $M_4$
coalesced with a singular point located on the hyperbola, whereas
the other one remains outside the hyperbola. Then, we obtain the
conditions and configurations as follow:
\begin{itemize}
 \item $\chi_C^{(3)}<0$ and $\chi_B^{(3)}<0$  $\Rightarrow$ Config. H.21;
 \item $\chi_C^{(3)}<0$ and $\chi_B^{(3)}>0$  $\Rightarrow$ Config. H.23;
 \item $\chi_C^{(3)}>0$ $\Rightarrow$ Config. H.20,
\end{itemize}
in which in the last case the condition $\chi_C^{(3)}>0$ implies
$\chi_B^{(3)}<0$, because ${\rm T}<0$ yields $0<g<1/2$ and, for these
values of $g$ combined with the condition $\chi_C^{(3)}>0$, we
have $a<0$ and hence $\chi_B^{(3)}<0$.


(ii.2) {\it Subcase ${\rm T}=0$.} In this case, by
conditions \eqref{cond:CS-4} and $\mu_0\ne0$, the equality
${\rm T}=0$ yields $g=3/5$ and hence $\chi_C^{(3)}=-416000/3<0$, which
leads to configuration given by Config. H.26.


($\alpha3$) {\it Possibility $\chi_A^{(3)}=0$.} From
\eqref{cond:CS-4},  condition $\chi_A^{(3)}=0$ implies
$1+4ag=0$ and hence $a=-1/(4g)$. In this case the points $M_{1,2}$
collapse and we have a double point on the hyperbola. In this case
we see that
$$
\chi_D^{(3)}=\frac{46(3g-1)^3(9g-1)^2}{81g^2}\ne0
$$
and $\delta_1\ne0$, by \eqref{cond:CS-4}. So, as
$\chi_A^{(3)}\ne0$ no other point could collapse with the double
point on the hyperbola, we arrive at the configuration
Config. H.7 if $\chi_B^{(3)}<0$ and
Config. H.8 if $\chi_B^{(3)}>0$.


($\beta$) {\it Subcase $\mu_0=0$.} We consider the
possibilities: $\chi_A^{(3)}<0$, $\chi_A^{(3)}>0$ and
$\chi_A^{(3)}=0$.


($\beta1$) {\it Possibility $\chi_A^{(3)}<0$.} The
singular points on the hyperbola are complex and, since $1+4ag<0$
yields $ag<0$, we have $g=2/3$ and then $a<0$, which is equivalent
to $\chi_B^{(3)}<0$. So we arrive at the configuration given by
Config. H.3.


($\beta2$) {\it Possibility $\chi_A^{(3)}>0$.}
Analogously we have $g=2/3$ and the points on the hyperbola are
real. We observe that,by  condition \eqref{cond:CS-4}, the
equality $\chi_C^{(3)}=0$ is equivalent to $g=0$. So we consider
two subcases: $\chi_C^{(3)}\ne0$ and $\chi_C^{(3)}=0$.


(i) {\it Case $\chi_C^{(3)}\ne0$.} Then one finite
singularity has gone to infinity and coalesced with the point
$[1,1,0]$. As observed earlier, this must be a singular point
located outside the hyperbola which goes to infinity and hence on
the finite part of the phase plane of systems \eqref{sys:CS-4-IL}
there are three singularities, two of which ($M_1$ and $M_2$)
being located on the hyperbola.

Since the singular points on the hyperbola are real, we have to
decide when the third point will belong also to the hyperbola.
For systems \eqref{sys:CS-4} with $g=2/3$ we calculate
\begin{gather*}
\chi_A^{(3)}=\frac{7713280}{81}\,(8a+3)(3a-2)^2,\quad
\chi_B^{(3)}=\frac{164798932}{81}\,a(3a-2)^2,\\
\chi_C^{(3)}=-\frac{133120}{3}\,a(3a-2)^2,\quad
\chi_D^{(3)}=\frac{736}{243}\,(a-3)(3a-2).
\end{gather*}
We observe that $\operatorname{sign}(\chi_B^{(3)})
=-\operatorname{sign}(\chi_C^{(3)})$ and, moreover,
$\chi_D^{(3)}=0$ (i.e. $a=3$) implies $\chi_C^{(3)}<0$.
So we get the following conditions and configurations:
\begin{itemize}
 \item $\chi_C^{(3)}<0$ and $\chi_D^{(3)}\ne0$  $\Rightarrow$  Config. H.32;
 \item $\chi_C^{(3)}<0$ and $\chi_D^{(3)}=0$  $\Rightarrow$  Config. H.34;
 \item $\chi_C^{(3)}>0$  $\Rightarrow$  Config. H.29.
\end{itemize}


(ii) {\it Case $\chi_C^{(3)}=0$.} Then $g=0$ and this implies
$$
\chi_B^{(3)}=164798932a(a+6)^2/81,\quad
\chi_C^{(3)}=0,\quad
\chi_D^{(3)}=-736(a_1)(a+6)^2/81.
$$
So we get the following conditions and configurations:
\begin{itemize}
 \item $\chi_B^{(3)}<0$ and $\chi_D^{(3)}\ne0$  $\Rightarrow$  Config. H.9;
 \item $\chi_B^{(3)}<0$ and $\chi_D^{(3)}=0$  $\Rightarrow$  Config. H.11;
 \item $\chi_B^{(3)}>0$  $\Rightarrow$  Config. H.10,
\end{itemize}
in which in the last case the condition $\chi_B^{(3)}>0$ (i.e.
$a>0$) implies $\chi_D^{(3)}\ne0$.


($\beta3$) {\it Possibility $\chi_A^{(3)}=0$.} Because of
 \eqref{cond:CS-4}, the condition $\mu_0=\chi_A^{(3)}=0$
implies $g(3g-2)=1+4ag=0$. Then this yields $g\ne0$ and hence
$g=2/3$ and $a=-3/8$. In this case the singularities $M_{1,2}$
coalesce and we have a double point on the hyperbola. For systems
\eqref{sys:CS-4} with $a=-3/8$ we calculate
$$
\chi_C^{(3)}=162500>0,\quad \chi_D^{(3)}=575/18\ne0.
$$
Since $\chi_D^{(3)}\ne0$, no other point could coalesce with the
double point on the hyperbola and we arrive at the configuration
Config. H.14.


(a2) {\it Case $B_1=0$.} Thus, according to
Lemma \ref{lem:Bi-0}, the condition $B_1=0$ is necessary in order
to exist an invariant line of systems \eqref{sys:CS-4}.
Considering \eqref{cond:CS-4}, the condition $B_1=0$ (see
\eqref{val:B1-CS4}) is equivalent to
$$
(2g-1)\big[3+a(3g+1)^2\big]=0.
$$
On the other hand for these systems we calculate
$$
\chi_E^{(3)}=(3g-1)\big[3+a(3g+1)^2\big] \big[6(1-3g) + a(3g+1)^2\big]
$$
and we examine two possibilities: $\chi_E^{(3)}\ne0$ and $\chi_E^{(3)}=0$.


($\alpha1$) {\it Subcase $\chi_E^{(3)}\ne0$.} In this
case we get $g=1/2$ and this leads to the systems
\begin{equation} \label{sys:CS-4-IL}
\frac {dx}{dt}= a+ x+ x^2/2-2xy/3,\quad
 \frac {dy}{dt}= - y (1+ x/2 - y/3),
\end{equation}
for which the following condition holds (see \eqref{cond:CS-4}):
\begin{equation} \label{cond:CS-4-IL}
a (25a-12) \ne0.
\end{equation}

Since the family of systems \eqref{sys:CS-4-IL} is a subfamily of
\eqref{sys:CS-4} (setting $g=1/2$), the invariant hyperbola
remains the same as in \eqref{Hyp:4}. Besides this hyperbola,
systems \eqref{sys:CS-4-IL} possess the invariant line $y=0$,
which is one of the asymptotes of this hyperbola. For the above
systems we calculate
\begin{gather*}
\mu_0=-1/36,\quad \chi_E^{(3)}=(25a+12)(25a-12)/192,\\
B_1=0,\quad B_2=-8a(25a+12) y^4.
\end{gather*}
Therefore, we conclude that, from conditions $\chi_E^{(3)}\ne0$ and
\eqref{cond:CS-4-IL}, we obtain $B_2\ne0$ and by Lemma \ref{lem:Bi-0} we could not
have another invariant line in a direction different from $y=0$.
Moreover, by  condition $\theta\ne0$ and according to
Lemma \ref{lem:theta,N}, in the direction $y=0$ we could neither
have a couple of parallel invariant lines nor a double invariant
line.

Since $\mu_0\ne0$, systems \eqref{sys:CS-4-IL} possess finite
singular points of multiplicity 4 with coordinates $M_i(x_i,y_i)$
($i=1,2,3,4$):
\begin{gather*}
x_{1,2}=-1\pm\sqrt{2a+1},\quad y_{1,2}=3(1\pm\sqrt{2a+1})/2,\\
x_{3,4}=-1\pm\sqrt{1-2a},\quad y_{3,4}=0.
\end{gather*}
We recall that the singular points $M_{1,2}$ are located on the
hyperbola and that the singularities $M_{3,4}$ are located on the
invariant line $y=0$.

On the other hand for systems \eqref{sys:CS-4-IL} we calculate
\begin{gather*}
\chi_A^{(3)}=\frac{482080}{243}(2a+1)(25a-12)^2,\quad
{\rm D}= a^2(2a-1)(2a+1)/3,\\
\chi_B^{(3)}=\frac{41199733}{5184}a(25a-12)^2,\quad
\chi_C^{(3)}=-\frac{2080}{9}a(25a-12)^2
\end{gather*}
and then the invariant polynomials $\chi_A^{(3)}$ and ${\rm D}$ govern
the types of the above singular points (i.e. are they real or
complex or coinciding), whereas the invariant polynomials
$\chi_B^{(3)}$ and $\chi_C^{(3)}$ are responsible respectively for
the position of the hyperbola and the location of the real
singularities on it (i.e. on the same branch or on the different
ones).


($\alpha1$) {\it Possibility $\chi_A^{(3)}<0$.}
Then the singularities $M_{1,2}$ (located on the hyperbola) are
complex. Since $\chi_A^{(3)}<0$, we obtain $\chi_B^{(3)}<0$ and
${\rm D}>0$, and by Proposition \ref{prop:number-FSPs} the
singularities on the invariant line are real and distinct. So we
get the configuration given by Config. H.49.


($\alpha2$) {\it Possibility $\chi_A^{(3)}>0$.} In
this case the singularities $M_{1,2}$ are real and they are
located on different branches (respectively on the same branch) of
the hyperbola if $\chi_C^{(3)}<0$ (respectively $\chi_C^{(3)}>0$).
We observe that the conditions $\chi_A^{(3)}>0$ and ${\rm D}\geq0$
imply $a\geq1/2$ and then $\chi_B^{(3)}>0$ and $\chi_C^{(3)}<0$.
Moreover, the conditions $\chi_A^{(3)}>0$ and $\chi_B^{(3)}<0$
yield $-1/2<a<0$ and then ${\rm D}<0$ and $\chi_C^{(3)}>0$. Therefore,
we arrive at the following conditions and configurations:
\begin{itemize}
 \item $\chi_B^{(3)}<0$  $\Rightarrow$  Config. H.74;
 \item $\chi_B^{(3)}>0$ and ${\rm D}<0$  $\Rightarrow$  Config. H.73;
 \item $\chi_B^{(3)}>0$ and ${\rm D}>0$  $\Rightarrow$  Config. H.47;
 \item $\chi_B^{(3)}>0$ and ${\rm D}=0$  $\Rightarrow$  Config. H.66.
\end{itemize}


($\alpha3$) {\it Possibility $\chi_A^{(3)}=0$.}
 By condition \eqref{cond:CS-4-IL}, the condition
$\chi_A^{(3)}=0$ implies $a=-1/2$. In this case the points
$M_{1,2}$ collapse and we have a double point on the hyperbola.
For systems \eqref{sys:CS-4-IL} with $a=-1/2$ we calculate
$$
\chi_B^{(3)}=41199733/41472>0,\quad
{\rm T}=-(3x-2y)^2 (9x^2-24xy+8y^2)^2/4478976<0.
$$
So, according to Proposition \ref{prop:number-FSPs}, besides the
double point on the hyperbola, we have two simple real singular
points on the invariant line $y=0$ and we get the configuration
given by Config. H.67.


($\beta$) {\it Subcase $\chi_E^{(3)}=0$.} In this
case we obtain $a=-3/(3g+1)^2$ and this leads to the systems
\begin{equation} \label{sys:CS-4-ILa}
\begin{gathered}
 \frac {dx}{dt}= -\frac{3}{(3g+1)^2}+ x+gx^2-2xy/3,\quad
 \frac {dy}{dt}= \frac{9(2g-1)}{(3g+1)^2}- y+(g-1)xy+y^2/3
\end{gathered}
\end{equation}
with the conditions
\begin{equation} \label{cond:CS-4-ILa}
 (g-1) (3g-1) (3g+1) (6g-1)\ne0.
\end{equation}
Moreover, systems \eqref{sys:CS-4-ILa} possess the following
invariant line and invariant hyperbola
\begin{equation} \label{hyp:H1+ILa}
 x-y +6/(3g+1)=0,\quad \Phi(x,y)=\frac{18}{(3g+1)^2}+2xy=0.
\end{equation}

We observe that the condition $\chi_E^{(3)}=0$ implies
\begin{gather*}
\chi_A^{(3)}=\frac{7713280(3g-1)^2 (6g-1)^2}{27(3g+1)^2}>0,\\
\chi_B^{(3)}=-\frac{164798932(g-1)^2 (3g-1)^2 (6g-1)^2}{3(3g+1)^2}<0,
\end{gather*}
because of \eqref{cond:CS-4-ILa}. Therefore, the points on the
hyperbola are real and distinct and the hyperbola assumes only
one position.

For the above systems we calculate
\begin{equation} \label{val:B2-CS4-ILa}
B_2= \frac{7776}{(3g+1)^4}(g-1)^2 (2g-1) (x-y)^4
\end{equation}
and, by Lemma \ref{lem:Bi-0}, for the existence of an invariant line
in a direction different from $y=x$ it is necessary $B_2=0$.


($\beta1$) {\it Possibility $B_2\ne0$.} Then
$2g-1\ne0$ and, since $\theta\ne0$, by Lemma \ref{lem:theta,N} we
could not have a couple of parallel invariant lines in the
direction $y=x$ and obviously the invariant line $y=x+6/(3g+1)$
is a simple one. As before, we consider two cases: $\mu_0\ne0$ and
$\mu_0=0$.


(i) {\it Case $\mu_0\ne0$.} Then $g(3g-2)\ne0$ and
systems \eqref{sys:CS-4-ILa} possess four real singularities
$M_i(x_i,y_i)$ having the following coordinates:
\begin{equation} \label{coord:FSPs-CS-4-ILa}
\begin{gathered}
x_1=-\frac{3}{3g+1},\quad y_1= \frac{3}{3g+1};\quad
x_2=-\frac{1}{g(3g+1)},\quad y_2= \frac{9g}{3g+1};\\
x_3=-\frac{1}{3g+1},\quad y_3= \frac{3(2g-1)}{3g+1};\\
x_4=-\frac{3}{(3g+1)(3g-2)},\quad y_4= \frac{9(2g-1)}{(3g+1)(3g-2)}.
\end{gathered}
\end{equation}

We could check directly that the singularity $M_1$ is a common
(tangency) point of the invariant hyperbola and of line
\eqref{hyp:H1+ILa}. Moreover, the singular point $M_2$
(respectively $M_4$) is located on the hyperbola (respectively on
the invariant line), whereas the singularity $M_3$ is generically
located outside the hyperbola as well as outside the invariant
line.

For systems \eqref{sys:CS-1-ILa} we calculate
\begin{equation} \label{val:D,chi-j,CS-4-ILa}
\begin{gathered}
 {\rm D}=-\frac{64}{3(3g+1)^8} (g+1)^2 (3g-5)^2 (3g-1)^6,\quad
 \mu_0=g(3g-2)/9,\\
 \chi_C^{(3)}=\frac{199680g(6g-1)^2}{(3g+1)^2},\quad
 \chi_D^{(3)}=\frac{1472(g+1)(3g-1)^2(6g-1)}{27(3g+1)^2}.
\end{gathered}
\end{equation}


(i.1) {\it Subcase $\chi_C^{(3)}<0$.} Then $g<0$
and the singular points $M_{1,2}$ are located on different
branches of the hyperbola and we obtain the configuration
Config. H.60 if ${\rm D}\ne0$ and Config. H.69 if
${\rm D}=0$.


(i.2) {\it Subcase $\chi_C^{(3)}>0$.} Then $g>0$
and the singular points $M_{1,2}$ are located on the same branch
of the hyperbola. It is clear that the reciprocal position of the
singularities $M_2$ (located on the hyperbola) and $M_4$ (located
on the invariant line) with respect to the tangency point $M_1$ of
the hyperbola and the invariant line \eqref{hyp:H1+ILa} defines
different configurations. More exactly, the type of the
configuration depends on the sign of the expression:
$$
(x_1-x_2)(x_1-x_4)=\frac{3(3g-1)^2}{g(3g-2) (3g+1)^2}.
$$
Hence, we observe that
$\operatorname{sign}\big((x_1-x_2)(x_1-x_4)\big)=\operatorname{sign}(\mu_0)$. So, if
${\rm D}\ne0$, we arrive at the configuration Config. H.61
if $\mu_0<0$ and Config. H.59 if $\mu_0>0$.

We consider now the case ${\rm D}=0$. Then, by condition
\eqref{sys:CS-1-ILa}, we have $(g+1)(3g-5)=0$ and clearly the
invariant polynomial $\chi_D^{(3)}$ distinguishes which one of the
two factors vanishes.

If $\chi_D^{(3)}\ne0$, then $g+1\ne0$ and we get $g=5/3$. We
observe that in this case the singularity $M_3$ collapses with
the singular point $M_4$ located on the invariant line. On the
other hand, we calculate
$$
{\rm T}=-256 (x-y)^2 (5x-y)^2 (5x+y)^2/177147<0
$$
and, by Proposition \ref{prop:number-FSPs}, we have three distinct
singularities (one of them being double). Now, assuming $g=5/3$,
for systems \eqref{sys:CS-4-ILa}, we calculate
$$
\chi_C^{(3)}=748800>0,\quad (x_1-x_2)(x_1-x_4)=4/15>0
$$
and hence we arrive at the configuration given by
Config. H.62.

In the case $\chi_D^{(3)}=0$, we have $g=-1$ and then the
singularity $M_3$ collapses with the singular point $M_2$ located
on the hyperbola (but outside of the invariant line). Moreover,
for $g=-1$ we have
$$
{\rm T}=-256 (x-y)^2 (3x+y)^2 (9x+y)^2/243<0
$$
and again we conclude that systems \eqref{sys:CS-4-ILa} possess
three distinct singularities (one double). In this case we have
$$
\chi_C^{(3)}=-2446080<0,\quad (x_1-x_2)(x_1-x_4)=12/5>0
$$
and therefore we get the configuration given by
Config. H.68.


(ii) {\it Case $\mu_0=0$.} Then $g(3g-2)=0$ and,
by Lemma \ref{lem:mu_i-ISPs}, at least one finite singularity has
gone to infinity and coalesced with an infinite singular point.
Since for systems \eqref{sys:CS-4-ILa} we have $\chi_C^{(3)}=0$ if
and only if $g=0$ (see \eqref{val:D,chi-j,CS-4-ILa}), we consider
two subcases: $\chi_C^{(3)}\ne0$ and $\chi_C^{(3)}=0$.


(ii.1) {\it Subcase $\chi_C^{(3)}\ne0$.} Then the
condition $\mu_0=0$ implies $3g-2=0$ (i.e. $g=2/3$) and,
considering the coordinates \eqref{coord:FSPs-CS-4-ILa} of the
finite singularities of systems \eqref{sys:CS-4-ILa}, we observe
that the singular point $M_4$ located on the invariant line has
gone to infinity and collapsed with the singularity $[1,1,0]$. In
this case calculation yields
$$
{\rm D}= -1600/19683<0,\quad \chi_C^{(3)}=133120>0,
$$
and, since by Remark \ref{rem:mu0=0,mu1-ne0} the condition
${\rm R}\ne0$ holds, according to Proposition \ref{prop:number-FSPs},
all three finite singularities are distinct. Moreover, becuase
$\chi_C^{(3)}>0$, the singularities are located on the same branch
of the hyperbola and we get the configuration given by
Config. H.57.


(ii.2) {\it Subcase $\chi_C^{(3)}=0$.} Then $g=0$
and in this case the singularity $M_2$ located on the hyperbola
\eqref{hyp:H1+ILa} has gone to infinity and coalesced with the
point $[1:0:0]$. Since by Remark \ref{rem:mu0=0,mu1-ne0} we have
$\mu_1\ne0$, according to Lemma \ref{lem:mu_i-ISPs} the other
three finite singular points remain on the finite part of the
phase plane.

Now, depending on the position of the singular point $M_4$
(located on the invariant line \eqref{hyp:H1+ILa}) with respect
to the vertical line $x=x_1$, we may get different configurations.
This distinction is governed by the sign of the expression
$x_4-x_1$ and we calculate
$$
{\rm D}=-1600/3\ne0,\quad x_4-x_1=3/2>0.
$$
Since by Remark \ref{rem:mu0=0,mu1-ne0} the condition ${\rm R}\ne0$
holds, according to Proposition \ref{prop:number-FSPs}, all three
finite singularities are distinct (${\rm D}\ne0$) and since
$x_4-x_1>0$, we arrive at the configuration given by
Config. H.50.


($\beta2$) {\it Possibility $B_2=0$.} Considering
\eqref{val:B2-CS4-ILa} and the condition \eqref{cond:CS-4}, we
obtain $g=1/2$ and this leads to the system
\begin{equation} \label{sys:CS-4-ILb}
\frac {dx}{dt}= -12/25+ x+ x^2/2- 2xy/3,\quad
 \frac {dy}{dt}= -y (1+ x/2 - y/3),
\end{equation}
possessing the two invariant lines and the invariant hyperbola:
$$
x-y+\frac{12}{5}=0,\quad y=0,\quad \Phi(x,y)=\frac{72}{25}+ 2 xy=0.
$$
as well as the following singularities $M_i(x_i,y_i)$ with the coordinates
\begin{equation} \label{coord:FSPs-CS-1-ILb2}
\begin{gathered}
x_1=-\frac{6}{5},\quad y_1= \frac{6}{5};\quad
x_2=-\frac{4}{5},\quad y_2= \frac{9}{5};\\
x_3= \frac{2}{5},\quad y_3= 0;\quad
x_4=-\frac{12}{5},\quad y_4=0.
\end{gathered}
\end{equation}

Hence, all singularities are located on the finite part of the
phase plane since $\mu_0=-1/36\ne0$. We calculate
$$
{\rm D}=-2352/390625<0,\quad \chi_C^{(3)}=319488/5>0.
$$

Since $\chi_C^{(3)}>0$, the singular points $M_1$ and $M_2$ are
located on the same branch of the hyperbola and we need to detect
the position of the singularity $M_2$ on the hyperbola. This fact
is verified by the sign of the expression
$(x_1-x_2)(x_1-x_4)=-12/25<0$. Then, we arrive at the
configuration given by Config. H.86.


(b) {\it Possibility $\delta_1=0$.} From condition
\eqref{cond:CS-4} we get $a=6(3g-1)/(3g+1)^2$ and we get the
following 1-parameter family of systems
\begin{equation} \label{sys:CS-4-delta1=0}
\begin{gathered}
 \frac {dx}{dt}=\frac{6(3g-1)}{(3g+1)^2}+ x+gx^2-\frac{2xy}{3},\\
 \frac {dy}{dt}=\frac{18(1-2g)(3g-1)}{(3g+1)^2}- y+(g-1)xy+\frac{y^2}{3},
\end{gathered}
\end{equation}
with the conditions
\begin{equation} \label{cond:CS-4-delta1=0}
 (g-1) (3g-1) (3g+1)\ne0.
\end{equation}

Moreover, systems \eqref{sys:CS-4-delta1=0} possess two invariant hyperbolas:
\begin{equation} \label{Hyp:4a}
\begin{gathered}
\Phi_1(x,y)=\frac{36(1-3g)}{(3g+1)^2}+2xy=0,\\
\Phi_2(x,y)=\frac{36(3g-1)}{(3g+1)^2}+\frac{12}{3g+1}x +2x(x-y)=0.
\end{gathered}
\end{equation}

We observe that the family of systems \eqref{sys:CS-4-delta1=0} is
a subfamily of systems \eqref{sys:CS-4} and hence, via the
transformation \eqref{transf:gamma4=0}, systems
\eqref{sys:CS-4-delta1=0} could be brought to systems of the same
form \eqref{sys:CS-4-delta1=0} but with the new parameter
$g_1=2/3-g$. So, this transformation induces a transformation in
the coefficient space which fixes the point $g=1/3$ and sends
the interval $(-\infty,1/3]$ onto the interval $[1/3,+\infty)$.
Thus, in what follows we shall consider only the values of the
parameter $g$ on the interval $(-\infty,1/3]$.

In this sense, we get the next remark.

\begin{remark} \label{rem:map} \rm
By an affine transformation and a time rescaling, we could
assume that the parameter $g$ in systems
\eqref{sys:CS-4-delta1=0} belongs to the interval $(-\infty,1/3]$.
\end{remark}

For systems \eqref{cond:CS-4-delta1=0} we calculate
\begin{equation} \label{val:B1-CS4-delta1=0}
 B_1=\frac{32}{(3g+1)^4} (g-1)^2 (3g-1)^3 (2g-1) (6g-1)
\end{equation}
and we analyze two subcases: $B_1\ne0$ and $B_1=0$.


(b1) {\it Case $B_1\ne0$.} In this case from
\eqref{cond:CS-4-delta1=0} we have $(2g-1)(6g-1)\ne0$. For
systems \eqref{sys:CS-4-delta1=0} we calculate $\mu_0=g(3g-2)/9$
and we consider two subcases: $\mu_0\ne0$ and $\mu_0=0$.


($\alpha$) {\it Subcase $\mu_0\ne0$.} Then the
systems have finite singularities of total multiplicity 4 with
coordinates:
\begin{equation} \label{coord:FSPs-CS-4-delta1=0}
\begin{gathered}
x_1=\frac{3g-1}{g(3g+1)},\quad y_1=\frac{18g}{3g+1};\quad
x_2=-\frac{6}{3g+1},\quad y_2=\frac{3(1-3g)}{3g+1};\\
x_3=\frac{3(3g-1)}{(3g+1)(3g-2)},\quad y_3= \frac{9(2g-1)(3g-1)}{(3g+1)(3g-2)};\\
x_4=-\frac{2}{3g+1},\ y_4=\frac{6(1-2g)}{3g+1}.
\end{gathered}
\end{equation}

We detect that the singularities $M_{1,2}$ are located on the
first invariant hyperbola \eqref{Hyp:4a} and moreover the
singularity $M_2$ is also located on the second hyperbola, i.e.
$M_2$ is a point of intersection of these two hyperbolas on the
finite part of the plane. The singular point $M_3$ belongs to
the second hyperbola, whereas the singularity $M_4$ generically is
located outside the hyperbolas.

For systems \eqref{Hyp:4a} we calculate
\begin{gather*}
\chi_F^{(3)}=(9g-1)(9g-5)/9,\quad \mu_0=g(3g-2)/9,\\
{\rm D}=-\frac{16}{3(3g+1)^8} (9g-1)^2(9g-5)^2 (5g-1)^2 (15g-7)^2.
\end{gather*}
On the other hand, we have
$$
x_1x_2=\frac{6(1-3g)}{g(3g+1)^2},\quad
\Phi_1(x_4,y_4)=\frac{12(1-5g)}{(3g-2)^2},\quad
\Phi_2(x_4,y_4)=\frac{4(15g-7)}{(3g-2)^2}.
$$

We observe that the singular points $M_{1,2}$ are located on
different branches (respectively on the same branch) of the first
hyperbola if only if $x_1x_2<0$ (respectively $x_1x_2>0$), and
this is governed by the $\operatorname{sign}(x_1x_2)
=-\operatorname{sign}\big(g(3g-1)\big)$.
Since by Remark \ref{rem:map} we have $g\in(-\infty,1/3]$, we
conclude that in this interval 
$\operatorname{sign}(x_1x_2)=-\operatorname{sign}(\mu_0)$.

Besides, we point out that the singular point $M_4(x_4,y_4)$
(which generically is located outside of the hyperbolas) could be
located on one of these invariant hyperbolas if and only if the
following condition holds:
$$
 \big[\Phi_1(M_4)\big] \big[\Phi_2(M_4)\big]
= [\frac{12(1-5g)}{(3g+1)^2}] [\frac{4(15g-7)}{(3g+1)^2}]
= \frac{48(1-5g)(15g-7)}{(3g+1)^4}=0.
$$
We observe that in the case $\chi_F^{(3)}\ne0$ the condition
$(5g-1)(15g-7)=0$ is equivalent to ${\rm D}=0$.


($\alpha1$) {\it Possibility $\mu_0<0$.} According
to Remark \ref{rem:map}, the condition $\mu_0<0$ is equivalent to
$g>0$ and the singular points $M_{1,2}$ are located on the same
branch of the first hyperbola. We calculate
$$
x_1-x_2=\frac{9g-1}{g(3g+1)}.
$$
We observe that $\operatorname{sign}(x_1-x_2)=\operatorname{sign}(\chi_F^{(3)})$ 
because of
 Remark \ref{rem:map}. Then we consider the cases $\chi_F^{(3)}<0$,
$\chi_F^{(3)}>0$ and $\chi_F^{(3)}=0$.


(i) {\it Case $\chi_F^{(3)}<0$.} Then
$(9g-1)(9g-5)<0$ and we consider two subcases: ${\rm D}\ne0$ and
${\rm D}=0$. If ${\rm D}\ne0$ we have only simple singular points on the
hyperbolas and we arrive at the configuration shown in
Config. H.128. Otherwise, ${\rm D}=0$ implies the existence
of a double singular point on the first hyperbola and this point
is characterized by the collision of the singular points $M_1$ and
$M_4$, and we get the configuration given by
Config. H.130.


(ii) {\it Case $\chi_F^{(3)}>0$.} Then
$(9g-1)(9g-5)>0$ and we get the configuration given by
Config. H.129.


(iii) {\it Case $\chi_F^{(3)}=0$.} Then
$(1-5g)(9g-5)=0$ and, according to Remark \ref{rem:map}, we get
$g=1/5$. In this case, the singularities $M_1$ and $M_2$ have
collided and we obtain a double singular point at the intersection
of the two hyperbolas \eqref{Hyp:4a} and hence we get the
configuration given by Config. H.124.


($\alpha2$) {\it Possibility $\mu_0>0$.} In this
case the singularities $M_{1,2}$ are located on different branches
of the first hyperbola and we get the configuration given by
Config. H.127.


($\beta$) {\it Subcase $\mu_0=0$.} Then $g=0$ and
the point $M_1$ has coalesced with the point $[1,0,0]$ at infinity
and we obtain the configuration shown in Config. H.125.


(b2) {\it Case $B_1=0$.} Considering
\eqref{cond:CS-4-delta1=0}, the condition $B_1=0$
(see \eqref{val:B1-CS4-delta1=0}) is equivalent to $(2g-1)(6g-1)=0$.
According to Remark \ref{rem:map}, we have $g=1/6$ and in this
case, besides the hyperbola, we have the invariant line $x-y+4=0$.
Since $B_2=-6400(x-y)^4/9\ne0$, the system could not possess
another invariant line by Lemma \ref{lem:Bi-0}. Moreover, we
observe that the point $M_1$ is the point of intersection of the
first hyperbola and the invariant line. Since $\mu_0=-1/36<0$ and
$\chi_{13}=-7/36<0$, we get the configuration given by
Config. H.135.

\subparagraph{Subcase $\beta_2=0$} Then $g=1/3$ and we
arrive at systems of the form
 \begin{equation} \label{sys:eta-poz-theta-ne0-gamma1=0c-beta2=0}
\frac {dx}{dt}= a+x+ x^2/3 -2xy/3,\quad
 \frac {dy}{dt}= b-y- 2xy/3+ y^2/3,
\end{equation}

For systems \eqref{sys:eta-poz-theta-ne0-gamma1=0c-beta2=0} we calculate
$$
\gamma_5=256ab(a-b)/81,\quad \mathcal{R}_4=128(a^2-ab+b^2)/6561.
$$
To have $\gamma_5=0$ we must have $ab(a-b)=0$. We observe
that in the case $ab=0$ we may assume $b=0$ from the change
$(x,y,t)\mapsto(-y,-x,-t)$. On the other hand, the systems
\eqref{sys:eta-poz-theta-ne0-gamma1=0c-beta2=0} with $b=0$ could
be brought to the same systems with $b=a$ via the change
$(x,y,t)\mapsto(x,x-y+3,-t)$. Therefore, we consider the
family of systems
 \begin{equation} \label{sys:CS-5}
\frac {dx}{dt}= -a/3+x+ x^2/3 -2xy/3,\quad
 \frac {dy}{dt}= -a/3-y- 2xy/3+ y^2/3,
\end{equation}
with the condition $a\ne0$.

We observe that the above family of systems is a subfamily of
systems \eqref{sys:CS-3} defined by the condition $h=1/3$. For
the family \eqref{sys:CS-3}, it was shown that, from
conditions \eqref{cond:CS-3} (i.e. $h\ne1/3$), we have
$\operatorname{sign}(\chi_A^{(2)})=\operatorname{sign}(1-4ah^2)$ and
$\operatorname{sign}(\chi_B^{(2)})=\operatorname{sign}(x_1x_2)$. 
Clearly that for the
subfamily \eqref{sys:CS-5} these invariants vanish and we need
other invariant polynomials which are responsible for the
$\operatorname{sign}(1-4ah^2)$ and $\operatorname{sign}(x_1x_2)$ 
in this particular case.

We calculate
$$
(1-4ah^2)\big|_{\{h=1/3\}}=(9-4a)/9,\quad (x_1x_2)\big|_{\{h=1/3\}}=a.
$$
On the other hand, for systems \eqref{sys:CS-5} we calculate
$$
\chi_A^{(3)}=123412480 a^2 (9-4a)/19683,\quad \chi_C^{(3)}=1064960a^3/729
$$
and hence $\operatorname{sign}(\chi_A^{(3)})=\operatorname{sign}(9-4a)$ 
and $\operatorname{sign}(\chi_C^{(3)})=\operatorname{sign}(x_1x_2)$.

Thus, considering the conditions and configurations for family
\eqref{sys:CS-3}, we get the configurations given by
Config. H.37 if $\chi_A^{(3)}<0$; Config. H.52
if $\chi_A^{(3)}>0$ and $\chi_C^{(3)}<0$; Config. H.53
if $\chi_A^{(3)}>0$ and $\chi_C^{(3)}>0$ and
Config. H.45 if $\chi_A^{(3)}=0$.

\paragraph{Case $\beta_6=0$.} The conditions $\beta_6=-c(g-1)(h-1)/2=0$ and
$\theta=(g-1) (h-1)(g+h)/2\ne0$ imply $c=0$. Then for systems
\eqref{sys:eta-poz-theta-ne0-gamma1=0} with $c=0$ we calculate
$$
\beta_7=2(2g-1) (2h-1) (1-2 g -2h), \quad
\gamma_5= -288(g-1)(h-1)(g+h)\mathcal{B}_1 \mathcal{B}_2\mathcal{B}_3,
$$
where
\begin{gather*}
\mathcal{B}_1\equiv b(2h-1)- a(2g-1);\quad
\mathcal{B}_2\equiv b(1-2h)+ 2a(g+2h-1);\\
\mathcal{B}_3 \equiv a(1-2g)+ 2b(2g+ h-1).
\end{gather*}
We consider two subcases: $\beta_7\ne0$ and $\beta_7=0$.

\begin{remark} \label{rem:g,h,1-g-h}
Considering systems \eqref{sys:eta-poz-theta-ne0-gamma1=0} with
$c=0$, having the relation $(2h-1)(2g-1)(1-2g-2h)=0$
(respectively $(4h-1)(4g-1)(3-4g-4h)=0$), by a change, we may
assume any of the factors $2h-1$, $2g-1$ or $1-2g-2h$
(respectively $4h-1$, $4g-1$ or $3-4g-4h$) to be zero, for
instance we could set $2h-1=0$ (respectively $4h-1=0$).
\end{remark}

Indeed, it is sufficient to observe that in the case $2g-1=0$
(respectively $4g-1=0$) we could apply the change
$$
(x,y,a,b,g,h)\mapsto(y,x,b,a,h,g),
$$
which conserves systems \eqref{sys:eta-poz-theta-ne0-gamma1=0}
with $c=0$, whereas in the case $1-2g-2h=0$ (respectively
$3-4g-4h =0$) we apply the change
$$
(x,y,a,b,g,h)\mapsto(y-x,-x,b-a,-a,h,1-g-h),
$$
which also conserves systems \eqref{sys:eta-poz-theta-ne0-gamma1=0} with $c=0$.

\subparagraph{Subcase $\beta_7\ne0$.} According to Theorem
\ref{trm:exist-hyp}, in this case for the existence of an
invariant hyperbola, it is necessary and sufficient $\gamma_5=0$,
which is equivalent to
$\mathcal{B}_1\mathcal{B}_2\mathcal{B}_3=0$. We claim that,
without loss of generality, we may assume $\mathcal{B}_1=0$, as
other cases could be brought to this one via an affine
transformation.

Indeed, assume first $\mathcal{B}_1\ne0$ and
$\mathcal{B}_2=0$. Then we apply to systems
\eqref{sys:eta-poz-theta-ne0-gamma1=0} with $c=0$ the linear transformation
$x'=y-x$, $y'=-x$ and we get the systems
$$
 \frac {dx'}{dt}= a'+g'x'^2+(h'-1)x'y',\quad
 \frac {dy'}{dt}= b'+(g'-1)x'y'+h'y'^2.
$$
These systems have the following new parameters:
$$
a'=b-a,\quad b'=-a,\quad g'=h, \quad h'=1-g-h.
$$
A straightforward computation gives
$$
\mathcal{B}_1'=b'(2h'-1)- a'(2g'-1)=b(1-2h) + 2 a (-1 + g + 2 h)=\mathcal{B}_2=0
$$
and hence, the condition $\mathcal{B}_2=0$ we replace by
$\mathcal{B}_1=0$ via a linear transformation.

Analogously in the case $\mathcal{B}_1\ne0$ and
$\mathcal{B}_3=0$, via the linear transformation  $x''=-y$, 
$y''= x-y$,  we replace the condition $\mathcal{B}_3=0$ by
$\mathcal{B}_1=0$ and this completes the proof of our claim.

Since $\beta_7\ne0$ (i.e. $2h-1\ne0$) the condition
$\mathcal{B}_1=0$ yields $b=a(2g-1)/(2h-1)$ and we arrive at
the 3-parameter family of systems
\begin{equation} \label{sys:CS-6}
\begin{gathered}
\frac {dx}{dt}= a(2h-1)+gx^2+(h-1)xy,\\
\frac {dy}{dt}= a(2g-1)+(g-1)xy+hy^2
\end{gathered}
\end{equation}
with the condition
\begin{equation} \label{cond:CS-6}
a (g-1) (h-1) (2g-1) (2h-1) (g+h) (2g+2h-1) \ne0.
\end{equation}
These systems possess the invariant hyperbola
\begin{equation} \label{Hyp:6}
\Phi(x,y)=a+xy=0.
\end{equation}
For systems \eqref{sys:CS-6} we calculate
\begin{equation} \label{val:B1-CS6}
\begin{gathered}
\beta_8=2 (4g-1) (4h-1) (3-4 g-4 h),\\
\begin{aligned}
\delta_2&=2a(4g-1)(4h-1)\big[68(g^2+h^2)+236gh-79(g+h)\\
&\quad -144gh(g+h)+22\big],\\
\end{aligned}
\end{gathered}
\end{equation}
According to Theorem \ref{trm:exist-hyp}, these systems possess
either one or two invariant hyperbolas if either
$\beta_8^2+\delta_2^2\ne0$ or $\beta_8=\delta_2=0$, respectively.

We claim that the condition $\beta_8=\delta_2=0$ is equivalent to
$(4g-1) (4h-1)=0$. Indeed, assuming that $(4g-1) (4h-1)\ne0$ and
$\beta_8=\delta_2=0$ we obtain
$$
3-4g-4h=0,\quad 68(g^2+h^2)+236gh-79(g+h)-144gh(g+h)+22=0.
$$
The first equation gives $g=3/4-h$ and then from the second one we
obtain $(2h-1)(4h-1)=0$, which contradicts the condition
\eqref{cond:CS-6} and the assumption. This completes the proof of
our claim.


(a) {\it Possibility $\beta_8^2+\delta_2^2\ne0$.}
Then this implies $(4g-1) (4h-1)\ne0$ and systems
\eqref{sys:CS-6} possess only one invariant hyperbola. For these
systems we calculate
$$
B_1= 2 a^3 (g-1)^2 (h-1)^2 (2g-1) (2h-1) (g-h) (g+h)^2
$$
and considering \eqref{cond:CS-6} we conclude that the condition
$B_1=0$ is equivalent to $g-h=0$. We examine two cases:
$B_1\ne0$ and $B_1=0$.


(a1) {\it Case $B_1\ne0$.} Then $g-h\ne0$ and
by Lemma \ref{lem:Bi-0} we have no invariant lines. For systems
\eqref{sys:CS-6} we calculate $\mu_0=gh(g+h-1)$ and we consider
two subcases: $\mu_0\ne0$ and $\mu_0=0$.


($\alpha$) {\it Subcase $\mu_0\ne0$.} In this case
the systems have finite singularities of total multiplicity 4
with the following coordinates $M_i(x_i,y_i)$:
%\label{page:x_i,y_i-CS6}
\begin{gather*}
x_{1,2}=\pm\frac{\sqrt{-agh}}{g},\quad y_{1,2}=\pm\frac{\sqrt{-agh}}{h},\\
x_{3,4}=\pm(2h-1)\frac{\sqrt{a(1-g-h)}}{g+h-1},\quad
y_{3,4}=\pm(2g-1) \frac{\sqrt{a(1-g-h)}}{g+h-1}.
\end{gather*}

We detect that the singularities $M_{1,2}$ are located on the
invariant hyperbola. More exactly, these singular points are
located on different branches (respectively on the same branch) of
the hyperbola if and only if $x_1x_2<0$ (respectively $x_1x_2>0$),
where $x_1x_2=ah/g$. Moreover, these singularities are real if
$agh<0$, they are complex if $agh>0$ and they coincide if $agh=0$.

On the other hand, we calculate
\begin{gather*}
\chi_A^{(4)}=-16128 a^5gh(g-1)^2(h-1)^2(g+h)^4(2g-1)^4(2h-1)^4(4g-1)^2(4h-1)^2,\\
\chi_B^{(4)}=-4257792 a^5(g+h)^4(2g-1)^6(2h-1)^6(4g-1)^2(4h-1)^2
\end{gather*}
and from the condition \eqref{cond:CS-6} we have
$\operatorname{sign}(\chi_A^{(4)})= -\operatorname{sign}(agh)
=-\operatorname{sign}(x_1x_2)$ and
$\operatorname{sign}(\chi_B^{(4)})=-\operatorname{sign}(a)$ 
(which corresponds to the position
of the hyperbola). We observe that in the case the singular points
$M_1$ and $M_2$ are real, they must be located on different
branches of the hyperbola (we recall that systems \eqref{sys:CS-6}
is symmetric with respect to the origin). Moreover, we could not
have $\chi_A^{(4)}=0$ because $\mu_0\ne0$ and \eqref{cond:CS-6}.

Besides, we point out that at least one of the singular points
$M_{3,4}$ could be located on the invariant hyperbola and we
determine the conditions for this to happen. We calculate
$$
 \Phi(x_3,y_3)=\Phi(x_4,y_4)=\frac{a(4gh-g-h)}{g+h-1}.
$$
It is clear that both of the singular points $M_3$ and $M_4$
belong to the hyperbola \eqref{Hyp:6} if and only if $4gh-g-h=0$.
Since
$$
{\rm D}=-768 a^4 (4gh-g-h)^4 \mu_0,
$$
we deduce that both of the singular points $M_{3,4}$ belong to the
hyperbola if and only if ${\rm D}=0$.


($\alpha1$) {\it Possibility $\chi_A^{(4)}<0$.} So
we have no real singularities located on the invariant hyperbolas
and we arrive at the configurations given by Config. H.1
if $\chi_B^{(4)}<0$ and Config. H.2 if $\chi_B^{(4)}>0$.


($\alpha2$) {\it Possibility $\chi_A^{(4)}>0$.} In
this case we have two real singularities located on the hyperbola
and they are located on different branches. Now, we need to
decide if both of the singular points $M_{3,4}$ will belong to the
hyperbola.


(i) {\it Case ${\rm D}\ne0$.} Then $4gh-g-h\ne0$ and
on the hyperbola there are two simple real singularities and we
obtain the configurations given by Config. H.17 if
$\chi_B^{(4)}<0$ and Config. H.19 if $\chi_B^{(4)}>0$.


(ii) {\it Case ${\rm D}=0$.} Then $4gh-g-h=0$ (i.e.
$g=h/(4h-1)$) and in this case we calculate
\begin{gather*}
\chi_A^{(4)}=-4128768\,a^5h^{10}(h-1)^2(2h-1)^8(3h-1h)^2/(4h-1)^{11},\\
{\rm D}={\rm T}=0,\quad {\rm P}{\rm R}=-256a^3h^8(2h-1)^8
\big[x-(4h-1)y\big]^6/(4h-1)^{11}
\end{gather*}
and, from $\chi_A^{(4)}>0$, we have ${\rm P}{\rm R}>0$ and on the
hyperbola there are two double real singularities (see Proposition
\ref{prop:number-FSPs}) we arrive at the configurations given by
Config. H.27 if $\chi_B^{(4)}<0$ and
Config. H.28 if $\chi_B^{(4)}>0$.


($\beta$) {\it Subcase $\mu_0=0$.} We consider the
possibilities: $\chi_A^{(4)}<0$, $\chi_A^{(4)}>0$ and
$\chi_A^{(4)}=0$.


($\beta1$) {\it Possibility $\chi_A^{(4)}<0$.} Then
$gh\ne0$ and the condition $\mu_0=0$ yields $g=1-h$. So we
calculate
$$
{\rm D}=0,\quad \mu_1=0,\quad \mu_2=a h (1-h) (2h-1)^2 (x-y)^2\ne0.
$$
Hence, two singular points go to infinity in the direction $y=x$
and we get the configurations Config. H.5 if
$\chi_B^{(4)}<0$ and Config. H.6 if $\chi_B^{(4)}>0$.


($\beta2$) {\it Possibility $\chi_A^{(4)}>0$.} As
in the previous subcase, two singular points go to infinity in the
direction $y=x$ and, moreover, the singularities $M_{1,2}$ are
real. So we obtain the configurations Config. H.35 if
$\chi_B^{(4)}<0$ and Config. H.36 if $\chi_B^{(4)}>0$.


(a2) {\it Case $B_1=0$.} Then by conditions
\eqref{cond:CS-6}, we get $g=h$ and systems
\eqref{sys:CS-6} possess the invariant line $x-y=0$. For this case
from \eqref{cond:CS-6} we have
$$
\mu_0=h^2(2h-1)\ne0,\quad {\rm D}=12288a^4h^6(1-2h)^5\ne0.
$$


($\alpha$) {\it Subcase $\chi_A^{(4)}<0$.} In this
case the singularities $M_{1,2}$ are complex and, since
$$
\chi_A^{(4)}=-258048\,a^5h^6(h-1)^4(2h-1)^8(4h-1)^4<0,
$$
we have $\chi_B^{(4)}=-68124672\,a^5h^4(2h-1)^{12}(4h-1)^4<0$. So,
we obtain the unique configuration Config. H.37.


($\beta$) {\it Subcase $\chi_A^{(4)}>0$.} In this
case the singularities $M_{1,2}$ are real and analogously we have
$\operatorname{sign}(\chi_A^{(4)})=\operatorname{sign}(\chi_B^{(4)})$. 
So we get the unique
configuration Config. H.53.


(b){\it Possibility $\beta_8=\delta_2=0$.} Then this
implies $(4g-1) (4h-1)=0$ and, by a change, we may assume
$h=1/4$, without loss of generality. In this case, systems
\eqref{sys:CS-6} possess the two invariant hyperbolas
$$
\Phi_1(x,y)=a+xy=0,\quad \Phi_2(x,y)=a-x(x-y)=0.
$$

For these systems we calculate
$$
\mu_0=g(4g-3)/16,\quad B_1= 9 a^3 (g-1)^2 (2g-1) (4g-1) (4g+1)^2/1024
$$
and, by conditions \eqref{cond:CS-6}, we verify that
$B_1\ne0$. Then we consider two cases $\mu_0\ne0$ and $\mu_0=0$.


(b1) {\it Case $\mu_0\ne0$.} Then
$g(4g-3)\ne0$ and the systems have finite singularities of total
multiplicity 4 with the following coordinates $M_i(x_i,y_i)$:
%\label{page:x_i,y_i-CS6}
\begin{gather*}
x_{1,2}=\pm\frac{\sqrt{-ag}}{2g},\quad
y_{1,2}=\pm2\sqrt{-ag},\\
x_{3,4}=\pm\frac{\sqrt{-a(4g-3)}}{4g-3},\quad y_{3,4}=\pm2(2g-1)
\frac{\sqrt{-a(4g-3)}}{4g-3}.
\end{gather*}

We detect that the singularities $M_{1,2}$ are located on the
invariant hyperbola $\Phi_1(x,y)=0$. More exactly, these singular
points are located on different branches (respectively on the same
branch) of the hyperbola if only if $x_1x_2<0$ (respectively
$x_1x_2>0$), where $x_1x_2=a/4g$. Moreover, these singularities
are real if $ag<0$, they are complex if $ag>0$ and they coincide
if $ag=0$. We also point out that the position of the hyperbolas
are governed by $\operatorname{sign}(a)$.

On the other hand, we calculate
$$
\chi_A^{(5)}=-41\,a(8g-3)^3/128.
$$

We observe that in the case the singular points $M_1$ and $M_2$
are real, they must be located on different branches of the
hyperbola (we recall that systems \eqref{sys:CS-6} is symmetric
with respect to the origin). Moreover, we could not have
$\chi_A^{(5)}=0$ because $\mu_0\ne0$ and \eqref{cond:CS-6}.

Moreover, we also detect that the singularities $M_{3,4}$ are
located on the invariant hyperbola $\Phi_2(x,y)=0$ and their
position regarding on which branch they are located is also governed
by $\operatorname{sign}(a)$ and they will be complex, real or coinciding
depending on the sign of the expression $a(4g-3)$ and hence the
sign of $\mu_0$ plays an important role in this analysis.

Besides, we point out that the singular points $M_{1,2}$ could not
be located on the hyperbola $\Phi_2(x,y)=0$ and, conversely,
$M_{3,4}$ could not be located on the hyperbola $\Phi_1(x,y)=0$,
since we have
$$
\Phi_2(x_{1,2},y_{1,2})=\frac{a}{4g}\ne0,\quad
\Phi_1(x_{3,4},y_{3,4})= \frac{a}{3-4g}\ne0,
$$
because of conditions \eqref{cond:CS-6}.

We consider the case $\mu_0<0$ (i.e. $0<g<3/4$). Then, for these
values of $g$, we have $8g-3<0$ and, independently of the sign of
$a$, we get the unique configuration Config. H.123.

In the case $\mu_0>0$, we obtain the configuration
Config. H.121 if $\chi_A^{(5)}<0$ and
Config. H.131 if $\chi_A^{(5)}>0$.


(b2) {\it Case $\mu_0=0$.} Then $g(4g-3)=0$
and depending on which one of these two factors vanishes, we
have different finite singular points coalescing with an infinite
singular point. More precisely, if $4g-3=0$ then the singular
points $M_{3,4}$ coalesce with $[1,1,0]$, and if $g=0$ then the
singular points $M_{1,2}$ coalesce with $[1,0,0]$.

However, we observe that, applying the change
$(x,y,t,a)\mapsto(-x,y-x,t,-a)$, we could bring systems
\eqref{cond:CS-6} with $h=1/4$ and $g=3/4$ to the same systems
with $h=1/4$ and $g=0$. So, without loss of generality, we may
assume $g=0$.

Thus, we obtain the configurations given by
Config. H.122 if $\chi_A^{(5)}<0$ and
Config. H.126 if $\chi_A^{(5)}>0$.

\subparagraph{Subcase $\beta_7=0$.} We recall that the
conditions $\beta_1=\beta_6=0$ yields $c=0$ and systems
\eqref{sys:eta-poz-theta-ne0-gamma1=0} with $c=0$ becomes
\begin{equation} \label{sys:eta-poz-theta-ne0-gamma1=0-beta1=beta6=0}
\frac {dx}{dt}= a+gx^2+(h-1)xy,\quad
 \frac {dy}{dt}= b+(g-1)xy+hy^2.
\end{equation}
Without loss of generality, Remark \ref{rem:g,h,1-g-h} assures us
that we may choose $g=1/2-h$ in order to have
$\beta_7=2(2g-1)(2h-1)(1-2g-2h)=0$.

Now, we calculate
$$
\beta_9=4h(1-2h)
$$
and we analyze two possibilities: $\beta_9\ne0$ and $\beta_9=0$.


(a) {\it Possibility $\beta_9\ne0$.} As earlier,
according to Theorem \ref{trm:exist-hyp}, in this case for the
existence of at least one invariant hyperbola, it is necessary and
sufficient $\gamma_5=0$, which is equivalent to
$\mathcal{B}_1\mathcal{B}_2\mathcal{B}_3=0$ and, without loss of
generality, we may assume $\mathcal{B}_1=0$, as other cases could
be brought to this one via an affine transformation.


(a1) {\it Case $\delta_3\ne0$.} In this case
we have only one invariant hyperbola and the condition
$\delta_3\ne0$ yields $a-b\ne0$. Then, the condition $\gamma_5=0$
is equivalent to $b(1-2h)-2ah=0$, which could be rewritten as
$a=a_1(2h-1)$ and $b=-2a_1h$. So, setting the old parameter $a$
instead of $a_1$, we arrive at the 2-parameter family of systems
\begin{equation} \label{sys:CS-7}
\begin{gathered}
\frac {dx}{dt}= a(2h-1)+(1-2h)x^2/2+(h-1)xy,\quad
\frac {dy}{dt}= -2ah-(2h+1)xy/2+hy^2,
\end{gathered}
\end{equation}
with the condition
\begin{equation} \label{cond:CS-7}
a h (h-1) (2h-1) (2h+1) (4h-1)\ne0.
\end{equation}
These systems possess the invariant hyperbola
\begin{equation} \label{Hyp:7}
\Phi(x,y)=a+xy=0.
\end{equation}
We observe that, from \eqref{cond:CS-7},
$$
B_1=a^3 h (h-1)^2 (2h-1) (2h+1)^2 (4h-1)\ne0
$$
and, hence, systems \eqref{sys:CS-7} possess no invariant line. Moreover, we have
$$
\mu_0=h(2h-1)/4\ne0,\quad {\rm D}=12 a^4 h (1-2h) (1-4h+8h^2)^4\ne0,
$$
because of the same conditions, and then all the finite singularities
remain in the finite part of the phase plane and none of them
coalesces with other points. Considering the coordinates of these
singularities $M_i(x_i,y_i)$ (i=1,2,3,4), we have
\begin{gather*}
x_{1,2}=\pm\frac{\sqrt{2ah(2h-1)}}{2h-1},\quad
y_{1,2}=\mp\frac{\sqrt{2ah(2h-1)}}{2h},\\
x_{3,4}=\pm(2h-1)\sqrt{2a},\quad y_{3,4}=\pm2h\sqrt{2a}.
\end{gather*}

After simple calculations, we obtain that $M_{1,2}$ are located on
the hyperbola, whereas $M_{3,4}$ are located generically outside
the hyperbola. Then, the singular points $M_{1,2}$ are complex if
$ah(2h-1)<0$ and they are real if $ah(2h-1)>0$. We point out that
these two singularities could not coincide since $ah(2h-1)\ne0$,
because \eqref{cond:CS-7}. So, we need to control
$\operatorname{sign}\big(ah(2h-1)\big)$. Moreover, $\operatorname{sign}(a)$ 
gives the position
of the hyperbola on the phase plane.

On the other hand, we calculate
\begin{gather*}
\chi_A^{(4)}=2016\,a^5 h^5 (h-1)^2 (2h-1)^5 (2h+1)^2 (4h-1)^4,\\
\chi_B^{(4)}=-17031168\,a^5 h^6 (2h-1)^6 (4h-1)^4.
\end{gather*}
Therefore, we arrive at the following conditions and configurations:
\begin{itemize}
 \item $\chi_A^{(4)}<0$ and $\chi_B^{(4)}<0$ $\Rightarrow$  Config. H.1;
 \item $\chi_A^{(4)}<0$ and $\chi_B^{(4)}>0$ $\Rightarrow$  Config. H.2;
 \item $\chi_A^{(4)}>0$ and $\chi_B^{(4)}<0$ $\Rightarrow$  Config. H.17;
 \item $\chi_A^{(4)}>0$ and $\chi_B^{(4)}>0$ $\Rightarrow$  Config. H.19.
\end{itemize}


(a2) {\it Case $\delta_3=0$.} In this case, the
conditions $\gamma_5=\delta_3=0$ yield $a-b=0$ (i.e. $b=a$) and
systems
\begin{equation} \label{sys:CS-8}
\frac {dx}{dt}= a+(1-2h)x^2/2+(h-1)xy,\quad
 \frac {dy}{dt}= a-(2h+1)xy/2+hy^2,
\end{equation}
with the condition
\begin{equation} \label{cond:CS-8}
a h (h-1) (2h-1) (2h+1)\ne0,
\end{equation}
possess at least two invariant hyperbolas. We calculate
$\beta_8= -2 (4h-1)^2$ and we analyze two subcases: $\beta_8\ne0$
and $\beta_8=0$.


($\alpha$) {\it Subcase $\beta_8\ne0$.} Then
$4h-1\ne0$ and systems \eqref{sys:CS-8} possess two invariant
hyperbolas:
\begin{equation} \label{Hyp:8}
\Phi_1(x,y)=-\frac{a}{2h-1}+x(x-y)=0,\quad
\Phi_2(x,y)=\frac{a}{h}+2y(x-y)=0.
\end{equation}
We observe that
$$
B_1=0, \quad B_2=-162 a^2 (h-1)^2 (2h+1)^2 (x-y)^4\ne0,
$$
because of \eqref{cond:CS-8}, and this implies that systems
\eqref{sys:CS-8} possess only one invariant straight line, namely
$x-y=0$.

From condition \eqref{cond:CS-8}, we obtain
$$
\mu_0=h(2h-1)/4\ne0,\quad {\rm D}=-12 a^4 h (2h-1)\ne0,
$$
and then we have four distinct finite singularities $M_i(x_i,y_i)$
($i=1,2,3,4$), where
\begin{gather*}
x_{1,2}=\pm\frac{\sqrt{2ah(2h-1)}}{2h-1},\quad
y_{1,2}=\pm\frac{\sqrt{2ah(2h-1)}}{2h},\\
x_{3,4}=\pm\sqrt{2a},\quad y_{3,4}=\pm\sqrt{2a}.
\end{gather*}

We observe that the singular points $M_{1,2}$ are located on the
first hyperbola \eqref{Hyp:8}, whereas $M_{3,4}$ are located on
the invariant line. Additionally, the singularities $M_{1,2}$
(respectively $M_{3,4}$) are complex if $ah(2h-1)<0$
(respectively $a<0$) and are real if $ah(2h-1)>0$ (respectively $a>0$).

So, we need to control $\operatorname{sign}\big(ah(2h-1)\big)$ and 
$\operatorname{sign}(a)$.
Moreover, $\operatorname{sign}\big(h(2h-1)\big)$ gives the position of the
hyperbolas on the phase plane.

On the other hand, we calculate
$$
\chi_A^{(6)}=ah(2h-1).
$$

If $\chi_A^{(6)}<0$, then the singularities $M_{1,2}$ are complex
and we get the configuration Config. H.132 if ${\rm D}<0$
and Config. H.133 if ${\rm D}>0$.

In the case $\chi_A^{(6)}>0$, the singular points $M_{1,2}$ are
real and we obtain the configuration Config. H.136 if
${\rm D}<0$ and Config. H.134 if ${\rm D}>0$.


($\beta$) {\it Subcase $\beta_8=0$.} Then $h=1/4$
and systems \eqref{sys:CS-8} possess three invariant hyperbolas,
namely the two presented in \eqref{Hyp:8} with $h=1/4$ and
$$
\Phi_3(x,y)=2a-xy=0.
$$

In this case, we observe that ${\rm D}=3a^4/2>0$ and we obtain the
configuration Config. H.156 if $\chi_A^{(6)}<0$ and
Config. H.157 if $\chi_A^{(6)}>0$.


(b) {\it Possibility $\beta_9=0$.} Then $h=0$ (this
yields $g=1/2$) and systems
\eqref{sys:eta-poz-theta-ne0-gamma1=0-beta1=beta6=0} becomes
\begin{equation} \label{sys:eta-poz-theta-ne0-gamma1=0-beta1=beta6=beta9=0}
\frac {dx}{dt}= a+x^2/2-xy,\quad
 \frac {dy}{dt}= b-xy/2.
\end{equation}

According to Theorem \ref{trm:exist-hyp}, in this case for the
existence of at least one invariant hyperbola, it is necessary and
sufficient $\gamma_6=0$, which is equivalent to $(a-b)b=0$.
Without loss of generality, we may assume $b=0$, since we could
pass from the case $b=a$ to the case $b=0$, via the affine
transformation $(x,y,t)\mapsto(x,x-y,-t)$. Then, we arrive at the
1-parameter family of systems
\begin{equation} \label{sys:CS-9}
\frac {dx}{dt}= a+x^2/2-xy,\quad  \frac {dy}{dt}= -xy/2.
\end{equation}
with the condition $a\ne0$.

The above family possesses the invariant hyperbola
\begin{equation} \label{Hyp:9}
\Phi(x,y)=a-xy=0
\end{equation}
and, since $B_1=0$ and $B_2=-162 a^2 y^4\ne0$, because $a\ne0$,
systems \eqref{sys:CS-9} possess the only one invariant line
$y=0$.

We calculate
$$
\mu_0=\mu_1=0,\quad \mu_2=ax^2/8,\quad {\rm D}=0.
$$
Then, two finite singular points has coalesced and have gone to
infinity and coalesced with $[0,1,0]$. Considering the remaining
singularities on the finite part of the plane, their coordinates
are $M_i(x_i,y_i)$ ($i=1,2$):
$$
x_{1,2}=\pm\sqrt{-2a},\quad
y_{1,2}=0.
$$

We point out that these two singularities are located on the
invariant line and they are complex if $a>0$ and are real if
$a<0$. So, we need to control $\operatorname{sign}(a)$, which also gives the
position of the hyperbola on the phase plane.

On the other hand, we calculate
$$
\chi_A^{(5)}=-a/16.
$$
So, we obtain the configuration Config. H.40 if
$\chi_A^{(5)}<0$ and Config. H.58 if $\chi_A^{(5)}>0$.

\subsection{Subcase $\theta=0$}

For systems \eqref{sys:QSgenCoef} we assume $\eta>0$ and therefore
we consider systems \eqref{sys:eta-poz-Gen} for which we have
$$
\theta=-(g-1)(h-1)(g+h)/2.
$$

Since $\theta=0$, we get $(g-1)(h-1)(g+h)=0$ and we may assume
$g=-h$, otherwise in the case $g=1$ (respectively $h=1$) we apply
the change $(x,y,g,h)\mapsto(-y,x-y,1-g- h,g)$ (respectively
$(x,y,g,h)\mapsto(y-x,-x,h,1-g-h)$) which preserves the quadratic
parts of systems~ \eqref{sys:eta-poz-Gen}.

So, $g=-h$ and we arrive at the systems
\begin{equation} \label{sys:eta-poz-theta=0}
\frac {dx}{dt}= a+cx-hx^2+(h-1)xy,\quad
 \frac {dy}{dt}= b+fy+(h+1)xy+hy^2,
\end{equation}
for which we calculate $N=9(h^2-1)(x-y)^2$. We consider two
possibilities: $N\ne0$ and $N=0$.

\subsubsection{Possibility $N\ne0$}

For systems \eqref{sys:eta-poz-theta=0}, we calculate
\begin{gather*}
\gamma_1=(c-f)^2 (c+f) (h-1)^2 (h+1)^2 (3h-1) (3h+1)/64,\\
\beta_6=(c-f)(h-1)(h+1)/4, \quad \beta_{10}=-2(3h-1)(3h+1).
\end{gather*}
According to Theorem \ref{trm:exist-hyp}, a necessary condition
for the existence of hyperbolas for these systems is
$\gamma_1=0$.

\paragraph{Case $\beta_6\ne0$.} Then $c-f\ne0$ and the condition
$\gamma_1=0$ yields $(c+f)(3h-1)(3h+1)=0$. So, we consider the
subcases: $\beta_{10}\ne0$ and $\beta_{10}=0$.

\subparagraph{Subcase $\beta_{10}\ne0$.} Then $(3h-1)(3h+1)\ne0$
and we get $f=-c$ and obtain the following systems
\begin{equation} \label{sys:eta-poz-theta=0-gamma1=0}
\begin{gathered}
\frac {dx}{dt}= a+cx-hx^2+(h-1)xy,\\
 \frac {dy}{dt}= b-cy-(h+1)xy+hy^2.
\end{gathered}
\end{equation}
Now,  to possess at least one hyperbola, it is necessary
and sufficient that for the above systems the condition
$$
\gamma_7=8 (h-1) (h+1) \big[a(2h+1)+b(2h-1)\big]=0
$$
holds, and because $N\ne0$ this is equivalent to
$a(2h+1)+b(2h-1)=0$.

Since $\beta_6=c(h-1)(h+1)/2\ne0$ (i.e. $c\ne0$), we could apply
the rescaling $(x,y,t)\mapsto(cx,cy,t/c)$ and assume $c=1$.
Moreover, since $(2h-1)^2+(2h+1)^2\ne0$, the condition
$a(2h+1)+b(2h-1)=0$ could be written as $a=-a_1(2h-1)$ and
$b=a_1(2h+1)$. So, setting the old parameter $a$ instead of $a_1$,
we arrive at the 2-parameter family of systems
\begin{equation} \label{sys:CSa-1}
\begin{gathered}
\frac {dx}{dt}= a(2h-1)+x-hx^2+(h-1)xy,\\
 \frac {dy}{dt}= -a(2h+1)-y-(h+1)xy+hy^2,
\end{gathered}
\end{equation}
with the condition
\begin{equation} \label{cond:CSa-1}
a(h-1)(h+1)(3h-1)(3h+1)\ne0.
\end{equation}
We observe that the family of systems \eqref{sys:CSa-1} is a
subfamily of systems \eqref{sys:CS-1} with $g=-h$.

The above systems possess the invariant hyperbola
\begin{equation} \label{Hyp:1a}
\Phi(x,y)=a+xy=0
\end{equation}
and for them we calculate
\begin{equation} \label{val:B1-CSa1}
B_1= -4 a^2 h (h-1)^2 (h+1)^2 (2h-1) (2h+1).
\end{equation}


(a) {\it Possibility $B_1\ne0$.} Then
$h(2h-1)(2h+1)\ne0$ and by Lemma \ref{lem:Bi-0} systems
\eqref{sys:CSa-1} possess no invariant lines. Since
$\mu_0=h^2\ne0$, these systems have finite singularities
$M_i(x_i,y_i)$ of total multiplicity 4, whose coordinates are
\begin{gather*}
x_{1,2}=\frac{1\pm\sqrt{1+4ah^2}}{2h},\quad
y_{1,2}=\frac{1\mp\sqrt{1+4ah^2}}{2h},\\
x_{3,4}=\frac{(2h-1)(1\pm\sqrt{1+4a})}{2},\quad
y_{3,4}=\frac{(2h+1)(1\pm\sqrt{1+4a})}{2}.
\end{gather*}

We observe that the singular points $M_{1,2}$ are located on the
hyperbola, whereas the singularities $M_{3,4}$ are generically
located outside of it.

On the other hand, for systems \eqref{sys:CSa-1}, we calculate the
invariant polynomials
\begin{gather*}
\chi_A^{(1)}=h^2(h-1)^2(h+1)^2(3h-1)^2(3h+1)^2 (1+4ah^2)/16,\\
\chi_B^{(1)}=-105\,ah^2(h-1)^2(h+1)^2(3h-1)^2(3h+1)^2/2
\end{gather*}
and, by the condition \eqref{cond:CSa-1}, we conclude that
$\operatorname{sign}(\chi_A^{(1)}) =\operatorname{sign}(1+4ah^2)$ 
(if $1+4ah^2\ne0$) and
$\operatorname{sign}(\chi_B^{(1)})=-\operatorname{sign}(a)$. 
So, we consider three cases:
$\chi_A^{(1)}<0$, $\chi_A^{(1)}>0$ and $\chi_A^{(1)}=0$.


(a1) {\it Case $\chi_A^{(1)}<0$.} Then
$1+4ah^2<0$ yields $a<0$ and hence $\chi_B^{(1)}>0$. So, since the
singular points located on the hyperbola are complex, we arrive at
the configuration given by Config. H.2.


(a2) {\it Case $\chi_A^{(1)}>0$.} In this case,
we have two real singularities located on the hyperbola. We
calculate $\ x_1x_2=-a $\ and, from the condition
\eqref{cond:CSa-1}, we obtain that
$\operatorname{sign}(\chi_B^{(1)})=\operatorname{sign}(x_1x_2)$, 
which defines the location of
the singular points $M_{1,2}$ concerning the branches of the
hyperbola (i.e. they are located either on different branches if
$\chi_B^{(1)}<0$ or on the same branch if $\chi_B^{(1)}>0$).

However, we need to detect when the singularities $M_{3,4}$ also
belong to the hyperbola. In this order, considering
\eqref{Hyp:1a}, we calculate
$$
\Phi(x,y)|_{\{x=x_{3,4},\ y=y_{3,4}\}}=
\frac{4h^2\big[1+2a\mp\sqrt{1+4a}\big]-1\pm\sqrt{1+4a}}{2}\equiv
\Omega_{3,4}(a,g,h).
$$
It is clear that at least one of the singular points $M_3$ or
$M_4$ belongs to the hyperbola \eqref{Hyp:1a} if and only if
$$
\Omega_3 \Omega_4= a (16ah^4+4h^2-1)=0.
$$
On the other hand, for systems \eqref{sys:CSa-1}, we have
$$
\chi_D^{(1)}=-105h(3h-1)(3h+1)(16ah^4+4h^2-1)
$$
and clearly, by \eqref{cond:CSa-1}, the condition
$\chi_D^{(1)}=0$ is equivalent to $16ah^4+4h^2-1=0$. We examine
two subcases: $\chi_D^{(1)}\ne0$ and $\chi_D^{(1)}=0$.


($\alpha$) {\it Subcase $\chi_D^{(1)}\ne0$.} Then,
on the hyperbola there only two simple real singularities and we
obtain the configurations given by Config. H.17 if
$\chi_B^{(1)}<0$ and Config. H.18 if $\chi_B^{(1)}>0$.


($\beta$) {\it Subcase $\chi_D^{(1)}=0$.} In this case, the condition
$16ah^4+4h^2-1=0$ yields $a=-(2h-1)(2h+1)/(16h^4)$ and we calculate
\begin{gather*}
{\rm D}=0,\\
\begin{aligned}
 {\rm T}&=-3(2h^2-1)^2 (x+y)^2 \big[(2h+1)x-(2h-1)y\big]^2 \\
&\quad\times \big[(h+1)(2h+1)x-(h-1)(2h-1)y\big]^2.
\end{aligned}
\end{gather*}
If ${\rm T}\ne0$, then we have a double and a simple singular points
on the hyperbola and we arrive at the configurations shown in
Config. H.21 if $\chi_B^{(1)}<0$ and
Config. H.22 if $\chi_B^{(1)}>0$. In the case ${\rm T}=0$,
we obtain $h=\pm\sqrt{2}/2$ and hence $\chi_B^{(1)}>0$. Then, we
have a triple and a simple singular points on the hyperbola and we
obtain the configuration Config. H.25.


(a3) {\it Case $\chi_A^{(1)}=0$.} Then
$a=-1/(4h^2)$ and hence $\chi_B^{(1)}>0$. In this case, the
singular points $M_1$ and $M_2$ coalesce and we get the
configuration Config. H.8.

(b) {\it Possibility $B_1=0$.} Then $h(2h-1)(2h+1)=0$ and we analyze
the two cases: $\mu_0\ne0$ and $\mu_0=0$.


(b1) {\it Case $\mu_0\ne0$.} Then $h\ne0$ and
the condition $B_1=0$ is equivalent to $(2h-1)(2h+1)=0$. Without
loss of generality, we may assume $h=-1/2$, otherwise we apply the
change $(x,y,t,h)\mapsto(-y,-x,-t,a,-h)$, which keeps systems
\eqref{sys:CSa-1} and changes the sign of $h$.

So $h=1/2$ and then systems \eqref{sys:CSa-1} possess the
invariant line $y=0$ and the singularities $M_{3,4}$ are located
on this line. In this case, we calculate
$$
\chi_A^{(1)}=225(a+1)/16384,\quad
\chi_B^{(1)}=-23625\,a/2048,\quad
{\rm D}=-48 a^2 (a+1) (4a+1).
$$


($\alpha$) {\it Subcase $\chi_A^{(1)}<0$.} Then
$a+1<0$ implies $a<0$ and hence $\chi_B^{(1)}>0$. So, we obtain
the configuration shown in Config. H.38.


($\beta$) {\it Subcase $\chi_A^{(1)}>0$.} Then $a>-1$ and we have real
singularities on the hyperbola. So, we get the following conditions and
configurations:
\begin{itemize}
 \item $\chi_B^{(1)}<0$ $\Rightarrow$  Config. H.75;
 \item $\chi_B^{(1)}>0$\ and\ ${\rm D}<0$ $\Rightarrow$  Config. H.72;
 \item $\chi_B^{(1)}>0$\ and\ ${\rm D}>0$ $\Rightarrow$  Config. H.46;
 \item $\chi_B^{(1)}>0$\ and\ ${\rm D}=0$ $\Rightarrow$  Config. H.65.
\end{itemize}


($\gamma$) {\it Subcase $\chi_A^{(1)}=0$.} Then
$a=-1$ (consequently ${\rm D}=0$ and $\chi_B^{(1)}>0$) and this
implies the existence of a double singular point on the hyperbola
and the singularities on the invariant line are complex, obtaining
the configuration Config. H.42.


(b2) {\it Case $\mu_0=0$.} Then $h=0$ and we also have $\mu_1=0$ and
$\mu_2=-xy$, which means that the singular points $M_{1,2}$ have gone
to infinity and coalesced with the singular points $[1,0,0]$ and $[0,1,0]$.

Considering Lemma \ref{lem:line-L} we detect that $Z$ is a simple
factor of $\mathcal{E}_1$ and $\mathcal{E}_2$. So, we deduce that
the infinity line $Z=0$ is a double invariant line for systems
\eqref{sys:CSa-1}. Since $\chi_A^{(1)}=1>0$, we obtain the
configurations Config. H.76 if $\chi_B^{(1)}<0$ and
Config. H.77 if $\chi_B^{(1)}>0$.

\subparagraph{Subcase $\beta_{10}=0$.} Then $(3h-1)(3h+1)=0$ and as earlier
we may assume $h=1/3$ and obtain the  systems
\begin{equation} \label{sys:CSa-2}
\frac {dx}{dt}= -\frac{a}{3}+x-\frac{x^2}{3}-\frac{2xy}{3},\quad
 \frac {dy}{dt}= -\frac{5a}{3}-y-\frac{4xy}{3}+\frac{y^2}{3},
\end{equation}
with the condition $a\ne0$. We again remark that the family of
systems \eqref{sys:CSa-2} is a subfamily of systems \eqref{sys:CS-1}
with $g=-h$ and $h=1/3$.

These systems possess the invariant hyperbola
\begin{equation} \label{Hyp:2a}
\Phi(x,y)=a+xy=0.
\end{equation}
and for them we calculate
$$
\mu_0=1/9,\quad {\rm D}=-16(4a+1)(4a+9)(16a-45)/19683,\quad B_1=1280a^2/2187.
$$
Since $B_1\ne0$, systems \eqref{sys:CSa-2} do not possess
invariant lines and the condition $\mu_0\ne0$ implies that the
finite singularities $M_i(x_i,y_i)$ are of total multiplicity 4,
and their coordinates are
\begin{gather*}
x_{1,2}=\frac{3\pm\sqrt{4a+9}}{2},\quad
y_{1,2}=\frac{3\mp\sqrt{4a+9}}{2},\\
x_{3,4}=\frac{-1\pm\sqrt{4a+1}}{6},\quad
y_{3,4}=\frac{5(1\mp\sqrt{4a+1})}{6}.
\end{gather*}

We observe that the singular points $M_{1,2}$ are located on the hyperbola,
 whereas the singularities $M_{3,4}$ are generically located outside of it.

Concerning the singular points $M_{1,2}$, we see that $x_1x_2=-a$ and
the $\operatorname{sign}(a)$ will detect the location of these singularities on
the same or different branches of the hyperbola as well as its position
on the phase plane.

Moreover, we need to detect when the singularities $M_{3,4}$ also belong
to the hyperbola. Considering \eqref{Hyp:2a}, we calculate
$$
\Phi(x,y)|_{\{x=x_{3,4},\ y=y_{3,4}\}}= \frac{8a\pm5\sqrt{4a+1}-5}{18}\equiv
\Omega'_{3,4}(a,g,h)
$$
and we observe that at least one of the singular points $M_3$ or
$M_4$ belongs to the hyperbola \eqref{Hyp:2a} if and only if
$$
\Omega'_3 \Omega'_4=\frac{a(16a-45)}{18}=0.
$$

On the other hand, for systems \eqref{sys:CSa-2}, we calculate the
invariant polynomials
\begin{gather*}
\chi_A^{(3)}=\frac{123412480(4a+9)}{243},\quad
\chi_B^{(3)}=-\frac{168754106368\,a}{243},\\
\chi_C^{(3)}=-\frac{1064960\,a}{9},\quad
\chi_D^{(3)}=\frac{5888(16a-45)}{729}
\end{gather*}
and we conclude that $\operatorname{sign}(\chi_A^{(3)})
=\operatorname{sign}(4a+9)$ (if
$4a+9\ne0$), $\operatorname{sign}(\chi_B^{(3)})
=\operatorname{sign}(\chi_C^{(3)})=-\operatorname{sign}(a)$
and at least one of the singular points $M_3$ or $M_4$ belongs to
the hyperbola if and only if $\chi_D^{(3)}=0$.

We observe that the condition $\chi_A^{(3)}<0$ implies
$\chi_B^{(3)}>0$ and $\chi_C^{(3)}>0$, all the finite singular
points are complex and we get the configuration
Config. H.2.

In the case $\chi_A^{(3)}>0$, the singularities $M_{1,2}$ are real
and we arrive at the following conditions and configurations:
\begin{itemize}
 \item $\chi_D^{(3)}\ne0$\ and\ $\chi_C^{(3)}<0$ $\Rightarrow$  Config. H.17;
 \item $\chi_D^{(3)}\ne0$\ and\ $\chi_C^{(3)}>0$ $\Rightarrow$  Config. H.18;
 \item $\chi_D^{(3)}=0$ $\Rightarrow$  Config. H.21.
\end{itemize}

And in the case $\chi_A^{(3)}=0$, the singular points $M_{1,2}$
have collapsed and $M_{3,4}$ are complex, obtaining the
configuration Config. H.8.

\paragraph{Case $\beta_6=0$.} Then $f=c$ and hence $\gamma_1=0$. We calculate
$$
\beta_2=c(h-1)(h+1)/2,\quad \beta_7=-2(2h-1)(2h+1)
$$
and we analyze two subcases: $\beta_2\ne0$ and $\beta_2=0$.

\subparagraph{Subcase $\beta_2\ne0$.} Then $c\ne0$ and we obtain the systems
\begin{equation} \label{sys:eta-poz-theta=0-gamma1=0-beta2ne0}
\frac {dx}{dt}= a+cx-hx^2+(h-1)xy,\quad
 \frac {dy}{dt}= b+cy-(h+1)xy+hy^2.
\end{equation}


(a) {\it Possibility $\beta_7\ne0$.} Then $(2h-1)(2h+1)\ne0$ and,
according to Theorem \ref{trm:exist-hyp}, for the existence of at
 least one invariant hyperbola for systems
\eqref{sys:eta-poz-theta=0-gamma1=0-beta2ne0},
it is necessary and sufficient the conditions $\gamma_8=0$ and
$\beta_{10}\mathcal{R}_7\ne0$. So, we calculate
\begin{gather*}
\gamma_8=42(h-1)(h+1)\,\mathcal{E}_2\mathcal{E}_3,\quad
 \beta_{10}=-2(3h-1)(3h+1),\\
\mathcal{E}_2=-2c^2(h-1)(2h-1)-2a(h-1)(3h-1)^2+b(2h-1)(3h-1)^2,\\
\mathcal{E}_3=-2c^2(h+1)(2h+1)+2b(h+1)(3h+1)^2-a(2h+1)(3h+1)^2.
\end{gather*}
We observe that the condition $\gamma_8=0$ is equivalent to
$\mathcal{E}_2\mathcal{E}_3=0$ and by the change
$(x,y,a,b,c,h)\mapsto(y,x,b,a,c,-h)$, we may assume that the
condition $\mathcal{E}_2=0$ holds.

Since $\beta_7\beta_{10}\ne0$, we could write the condition
$\mathcal{E}_2=0$ as $c=c_1(3h-1)$, $b=b_1(h-1)$ and
$a=(b_1-2c_1^2)(2h-1)/2$. Then, we apply the reparametrization
 $b_1=ac_1^2$ and $a=2a_1$.
Finally, since $c_1\ne0$ (because $c\ne0$), we could apply the
rescaling $(x,y,t)\mapsto(c_1x,c_1y,t/c_1)$ and assume $c_1=1$.
Thus, setting the old parameter $a$ instead of $a_1$, we arrive
at the 2-parameter family of systems
\begin{equation} \label{sys:CSa-3}
\begin{gathered}
 \frac {dx}{dt}= (a-1)(2h-1)+(3h-1)x-hx^2+(h-1)xy,\\
 \frac {dy}{dt}= 2a(h-1)+(3h-1)y-(h+1)xy+hy^2,
\end{gathered}
\end{equation}
with the conditions
\begin{equation} \label{cond:CSa-3}
(a-1)(h-1)(h+1)(2h-1)(2h+1)(3h-1)(3h+1)\ne0.
\end{equation}
These systems possess a couple of parallel invariant lines and
an invariant hyperbola:
\begin{equation} \label{Hyp:3a}
\begin{gathered}
\mathcal{L}_{1,2}(x,y)=h(x-y)^2-(3h-1)(x-y)+2h-a-1=0,\\
\Phi(x,y)=1-a-2x+x(x-y)=0.
\end{gathered}
\end{equation}

We remark that, since
$$
\operatorname{Discriminant}\big[\mathcal{L}_{1,2}(x,y),x-y\big]=(h-1)^2+4ah,
$$
these lines are complex (respectively real) if $(h-1)^2+4ah<0$
(respectively $(h-1)^2+4ah>0$).

We calculate
$$
\delta_4=3(h-1) (2h-1) \big[(h-1)^2(2h+1)+a(3h+1)^2\big]/2
$$
and we consider two cases: $\delta_4\ne0$ and $\delta_4=0$.

(a1) {\it Case $\delta_4\ne0$.} In this case we have $(h-1)^2(2h+1)+a(3h+1)^2\ne0$
and hence $\Phi(x,y)=0$ (see \eqref{Hyp:3a}) is the unique invariant hyperbola.
 Since $B_1=0$ for systems \eqref{sys:CSa-3}, we calculate
$$
B_2=-1296\,a (a-1) (h-1)^3 (h+1)^2 (2h-1) (x-y)^4.
$$


($\alpha$) {\it Subcase $B_2\ne0$.} Then $a\ne0$ and, since $\mu_0=h^2$,
we consider two possibilities: $\mu_0\ne0$ and $\mu_0=0$.


($\alpha1$) {\it Possibility $\mu_0\ne0$.} So we get $h\ne0$ and the
finite singularities of systems \eqref{sys:CSa-3} are of multiplicity 4,
and their coordinates are $M_i(x_i,y_i)$:
\begin{gather*}
x_{1,2}=\frac{h+1\pm\sqrt{(h-1)^2+4ah}}{2},\quad
y_{1,2}=\frac{(h-1)\big[h-1\pm\sqrt{(h-1)^2+4ah}\big]}{2h},\\
x_{3,4}=\frac{(2h-1)\big[h+1\pm\sqrt{(h-1)^2+4ah}\big]}{2h},\quad
y_{3,4}=h-1\pm\sqrt{(h-1)^2+4ah}.
\end{gather*}

We observe that the singular points $M_{1,2}$ are located on the hyperbola
and on the invariant lines, whereas the singularities $M_{3,4}$
are located on the invariant lines.

Concerning the singular points $M_{1,2}$, we see that $x_1x_2=h(1-a)$
and hence $\operatorname{sign}\big(h(a-1)\big)$ detects the location 
of these singularities
on the same or different branches of the hyperbola. Moreover,
the position of the hyperbola is governed by $\operatorname{sign}(a-1)$.

To detect when the singularities $M_{3,4}$ also belong to the hyperbola,
we consider \eqref{Hyp:3a} and we calculate
\begin{align*}
\Phi(x,y)|_{\{x=x_{3,4},\, y=y_{3,4}\}}
&= \frac{\tilde{A}\pm\big[(h+1)(2h-1) \sqrt{(h-1)^2+4ah}\big]}{2h^2} \\
&\equiv \Omega''_{3,4}(a,g,h)
\end{align*}
where $\tilde{A}=2ah(1-3h)+(1-h)(1-h+2h^2)$, and we observe that at least
one of the singular points $M_3$ or $M_4$ belongs to the hyperbola
\eqref{Hyp:3a} if and only if
$$
\Omega''_3 \Omega''_4=\frac{(a-1)\big[a(3h-1)^2+2(h-1)^3\big]}{h^2}=0.
$$
On the other hand, for systems \eqref{sys:CSa-3}, we calculate the
invariant polynomials
\begin{gather*}
\chi_A^{(7)}=(h-1)^2 (h+1)^2 \big[(h-1)^2+4ah\big]/16,\\
\chi_B^{(7)}=6480(a-1)(h-1)^2\big[(h-1)^2(2h+1)+a(3h+1)^2\big]^2,\\
\chi_C^{(7)}=2160\,h(1-a)(h-1)^2\big[(h-1)^2(2h+1)+a(3h+1)^2\big]^2,
\end{gather*}
and we conclude that $\operatorname{sign}(\chi_A^{(7)})
=\operatorname{sign}\big((h-1)^2+4ah\big)$
 (if $(h-1)^2+4ah\ne0$), $\operatorname{sign}(\chi_B^{(7)})
=\operatorname{sign}(a-1)$,
$\operatorname{sign}(\chi_C^{(7)})=\operatorname{sign}\big(h(1-a)\big)$ 
and at least one of the
singular points $M_3$ or $M_4$ belongs to the hyperbola if and only
if $a(3h-1)^2+2(h-1)^3=0$.


(i) {\it Case $\chi_A^{(7)}<0$.} Then all the
finite singular points are complex as well as the pair of
invariant lines. Moreover, the condition $\chi_A^{(7)}<0$ (i.e.
$(h-1)^2+4ah<0$) yields $ah<0$. Combining this inequality with
$\chi_B^{(7)}<0$ (i.e. $a-1<0$) (respectively $\chi_B^{(7)}>0$
(i.e. $a-1>0$)), we obtain $h<0$ (respectively $h>0$) and hence
$\chi_C^{(7)}<0$ (respectively $\chi_C^{(7)}>0$). So, we arrive at
the configuration Config. H.78 if $\chi_B^{(7)}<0$ and
Config. H.79 if $\chi_B^{(7)}>0$.


(ii) {\it Case $\chi_A^{(7)}>0$.}
Then all the finite singular points and the pair of invariant lines are real.
 In this sense, according to the position of the finite singular points
on the hyperbola and on the invariant lines, we may have different
configurations.

We calculate
\begin{gather*}
(x_1-x_4)(x_2-x_3)=-\frac{a(3h-1)^2+2(h-1)^3}{h},\\
(x_1-x_4)-(x_2-x_3)=\frac{(3h-1)\sqrt{(h-1)^2+4ah}}{h},\\
(x_1-x_4)+(x_2-x_3)=\frac{(1-h)(h+1)}{h}
\end{gather*}
and we observe that $\operatorname{sign}\big((x_1-x_4)(x_2-x_3)\big)$,
 $\operatorname{sign}\big((x_1-x_4)-(x_2-x_3)\big)$ and
$\operatorname{sign}\big((x_1-x_4)+(x_2-x_3)\big)$ govern the position of the four
finite singularities along the hyperbola and the invariant lines.
 More exactly, if $(x_1-x_4)(x_2-x_3)<0$ (respectively $(x_1-x_4)(x_2-x_3)>0$),
then the $\operatorname{sign}\big((x_1-x_4)-(x_2-x_3)\big)$
(respectively $\operatorname{sign}\big((x_1-x_4)+(x_2-x_3)\big)$)
distinguishes the position of $M_3$ and $M_4$ with respect to the hyperbola.

On the other hand, we calculate
\begin{gather*}
\chi_D^{(7)}=3(h-1)^2(h+1)^2\big[a(3h-1)^2+2(h-1)^3\big]/8,\\
\beta_{10}=-2(3h-1)(3h+1),\quad
N=9(h-1)(h+1)(x-y)^2.
\end{gather*}
We consider two subcases: $\chi_D^{(7)}\ne0$ and $\chi_D^{(7)}=0$.

(ii.1) {\it Subcase $\chi_D^{(7)}\ne0$.} In this case the singularities
 $M_{3,4}$ do not belong to the hyperbola and we need to distinguish when
 the singular points $M_{1,2}$ are located on different or on the same branch.


(ii.1.1) {\it Possibility $\chi_C^{(7)}<0$.} Then
$M_{1,2}$ are located on different branches of the hyperbola and,
if $\chi_B^{(7)}<0$, we obtain $a<0$ and $h<0$, and hence
$\chi_D^{(7)}<0$. So, we get the configuration
Config. H.96.

In the case $\chi_B^{(7)}>0$, we observe that the condition
$\chi_D^{(7)}<0$ implies $N<0$. So, we arrive at the following conditions
 and configurations:
\begin{itemize}
 \item $\chi_D^{(7)}<0$ $\Rightarrow$  Config. H.99;
 \item $\chi_D^{(7)}>0$ and $\beta_{10}<0$ $\Rightarrow$  Config. H.95;
 \item $\chi_D^{(7)}>0$ and $\beta_{10}>0$ $\Rightarrow$  Config. H.94.
\end{itemize}


(ii.1.2) {\it Possibility $\chi_C^{(7)}>0$.} Then $M_{1,2}$ are located
on the same branch of the hyperbola.

If $\chi_B^{(7)}<0$, the condition $\chi_D^{(7)}>0$ implies $\beta_{10}<0$
and we obtain the following conditions and configurations:
\begin{itemize}
 \item $\chi_D^{(7)}<0$ and $N<0$ $\Rightarrow$  Config. H.100;
 \item $\chi_D^{(7)}<0$ and $N>0$ $\Rightarrow$  Config. H.98;
 \item $\chi_D^{(7)}>0$ $\Rightarrow$  Config. H.97.
\end{itemize}

In the case $\chi_B^{(7)}>0$, the condition $\chi_D^{(7)}<0$
implies $\beta_{10}<0$. Moreover, if $\chi_D^{(7)}>0$,
independently of $\operatorname{sign}(N)$, we are led to the same configuration.
So, considering the claim stated in the next paragraph, we arrive
at the configuration Config. H.93 if $\chi_D^{(7)}<0$
and Config. H.92 if $\chi_D^{(7)}>0$.

We claim that, if $\chi_C^{(7)}>0$ and $\chi_B^{(7)}>0$ (i.e. the
 singular points $M_{1,2}$ are located on the
same branch and the hyperbola is positioned in the sense of
$\chi_B^{(7)}>0$), we could not have the configuration with the
singular points $M_{3,4}$ located inside the region delimited by
both branches of the hyperbola.

Indeed, suppose the contrary, that this configuration is
realizable. Then the conditions $\chi_A^{(7)}>0$, $\chi_B^{(7)}>0$
and $\chi_C^{(7)}>0$ are necessary and these conditions are
equivalent to
$$
(h-1)^2+4ah>0,\quad a-1>0, \quad h<0.
$$
We assume that $M_3$ and $M_4$ are located inside the region
delimited by both branches of the hyperbola. We observe that
inside this region we also have the origin of coordinates
(because $\Phi(0,0)=1-a<0$). Therefore we must have $\Omega''_3
\Omega''_4>0$ and
$\operatorname{sign}(\Omega''_3+\Omega''_4)
=\operatorname{sign}(\tilde{A})=\operatorname{sign}(1-a)$.
Hence the condition $\tilde{A}<0$ must hold. However, the
conditions $(h-1)^2+4ah>0$ and $h<0$ imply
$$
\tilde{A}=2ah(1-3h)+(1-h)(1-h+2h^2)\equiv \frac{1}{2}\big[(1-3h)
[(h-1)^2+4ah]+(1-h)(h+1)^2\big]>0,
$$
and this proves our claim.


(ii.2) {\it Subcase $\chi_D^{(7)}=0$.} Then
$a=-2(h-1)^3/(3h-1)^2$ and the singular points $M_4$ coalesces
with the singularity $M_1$. We note that the hyperbola divides
the plane into three regions: $\Phi(x,y)<0$, $\Phi(x,y)>0$ and
$\Phi(x,y)=0$, and the singular point $M_3$ could be located only
in the first two regions. Moreover,
$$
\Phi(M_3)=-\frac{(2h-1)(h-1)(h+1)^2}{h^2(3h-1)}
$$
and, in this case, we have
$$
\mathcal{L}_1=x-y+\frac{3h-1-4h^2}{h(3h-1)}=0,\quad
\mathcal{L}_2=x-y+\frac{3-5h}{3h-1}=0.
$$

We calculate
\begin{gather*}
\chi_A^{(7)}=(h-1)^4(h+1)^4/(16(3h-1)^2),\\
\chi_B^{(7)}=-58320(2h-1)(h-1)^6(h+1)^6/(3h-1)^6,\\
\chi_C^{(7)}=19440\,h(2h-1)(h-1)^6(h+1)^6/(3h-1)^6,\\
N=9(h-1)(h+1)(x-y)^2.
\end{gather*}
From condition \eqref{cond:CSa-3}, we have $\chi_A^{(7)}>0$,
$\operatorname{sign}(\chi_B^{(7)})=-\operatorname{sign}(2h-1)$,
$\operatorname{sign}(\chi_C^{(7)})=-\operatorname{sign}\big(h(2h-1)\big)$ and
$\operatorname{sign}(N)=-\operatorname{sign}\big((h-1)(h+1)\big)$. Moreover,
$\mathcal{L}_1-\mathcal{L}_2=(h-1)(h+1)/\big[h(3h-1)\big]$.

If $\chi_B^{(7)}<0$ (i.e $h>1/2$), we have $\chi_C^{(7)}>0$ and
$\operatorname{sign}\big(\Phi(M_3)\big)
=-\operatorname{sign}(\mathcal{L}_1-\mathcal{L}_2)=-\operatorname{sign}(N)$.
Then we get the configuration Config. H.89 if $N<0$ and
Config. H.90 if $N>0$.

In the case $\chi_B^{(7)}>0$ (i.e $h<1/2$), the condition
$\chi_C^{(7)}<0$ implies $N<0$ (then $x_2-x_3<0$), obtaining the
configuration Config. H.88. If $\chi_C^{(7)}>0$ (then
$\Phi(M_3)>0$), independently of the sign of $N$, we get the
configuration Config. H.87.


(iii) {\it Case $\chi_A^{(7)}=0$.} Then we have two double singular points 
(namely $M_1=M_2$ and $M_3=M_4$) and a double invariant line. 
The condition $\chi_A^{(7)}=0$ yields $a=-(h-1)^2/(4h)$ and hence 
$\chi_C^{(7)}>0$ and $\operatorname{sign}(\chi_B^{(7)})
=\operatorname{sign}(\chi_D^{(7)})=-\operatorname{sign}(h)$.

We observe that, if $\chi_B^{(7)}>0$, independently of 
$\operatorname{sign}(\beta_{10})$ and $\operatorname{sign}(N)$, 
we are conducted to the same configuration. Thus, we get the following 
conditions and configurations:
\begin{itemize}
 \item $\chi_B^{(7)}<0$ and $N<0$ $\Rightarrow$  Config. H.103;
 \item $\chi_B^{(7)}<0$ and $N>0$ $\Rightarrow$  Config. H.102;
 \item $\chi_B^{(7)}>0$ $\Rightarrow$  Config. H.101.
\end{itemize}

($\alpha2$) {\it Possibility $\mu_0=0$.} Then $h=0$ and, since we
 also obtain $\mu_1=0$ and $\mu_2=xy$, two finite singularities of
 systems \eqref{sys:CSa-3} have gone to infinity and coalesced with
$[1:0:0]$ and $[0:1:0]$. The remaining two finite singularities have
the coordinates $M_i(x_i,y_i)$:
$$
x_1=1, \quad y_1=-a,\quad
x_2=a-1, \quad y_2=-2.
$$
In this case, the invariant hyperbola remains the same, whereas one of
the invariant lines \eqref{Hyp:3a} goes to infinity and hence the
line of infinity $Z=0$ becomes double (see Lemma \ref{lem:line-L}).
The remaining invariant line is expressed by $x-y-(a+1)=0$.

We observe that the singular point $M_1$ is the intersection of the
hyperbola and the straight line, whereas $M_2$ is generically located
on the line and outside the hyperbola.
However, $M_2$ can be located on the hyperbola if and only if
$$
\Phi(x_2,y_2)=(a-1)(a-2)=0,
$$
which is possible if and only if $a-2=0$, because of conditions \eqref{cond:CSa-3}.

For systems \eqref{sys:CSa-3} with $h=0$, we calculate
$$
\chi_B^{(7)}=6480(a-1)(a+1)^2, \quad \chi_D^{(7)}=3(a-2)/8.
$$

We note that, if $\chi_B^{(7)}<0$, then $a<1$ and hence $\chi_D^{(7)}<0$.
So, we have the following conditions and configurations:
\begin{itemize}
 \item $\chi_B^{(7)}<0$ $\Rightarrow$  Config. H.106;
 \item $\chi_B^{(7)}>0$\ and\ $\chi_D^{(7)}<0$ $\Rightarrow$  Config. H.105;
 \item $\chi_B^{(7)}>0$\ and\ $\chi_D^{(7)}>0$ $\Rightarrow$  Config. H.107;
 \item $\chi_B^{(7)}>0$\ and\ $\chi_D^{(7)}=0$ $\Rightarrow$  Config. H.104.
\end{itemize}


($\beta$) {\it Subcase $B_2=0$.} Then $a=0$ and we arrive at the family of systems
\begin{equation} \label{sys:CSa-3-3IL}
\begin{gathered}
\frac {dx}{dt}= 1-2h+(3h-1)x-hx^2+(h-1)xy,\\
 \frac {dy}{dt}= (3h-1)y-(h+1)xy+hy^2,
\end{gathered}
\end{equation}
with the condition
\begin{equation} \label{cond:CSa-3-3IL}
(h-1)(h+1)(2h-1)(2h+1)(3h-1)(3h+1)\ne0.
\end{equation}
These systems possess three invariant lines and an invariant hyperbola
\begin{equation} \label{Hyp:3a-3IL}
\begin{gathered}
\mathcal{L}_1(x,y)=x-y-1=0,\quad
\mathcal{L}_2(x,y)=h(x-y)+1-2h=0,\\
\mathcal{L}_3(x,y)=y=0,\quad \Phi(x,y)=1-2x+x(x-y)=0.
\end{gathered}
\end{equation}

Since $\mu_0=h^2$, we consider again the possibilities:
$\mu_0\ne0$ and $\mu_0=0$.


($\beta1$) {\it Possibility $\mu_0\ne0$.} Then $h\ne0$ and the finite
singularities of systems \eqref{sys:CSa-3-3IL} are of multiplicity 4,
and their coordinates are $M_i(x_i,y_i)$:
\begin{gather*}
x_1=1, \quad y_1=0,\quad
x_2=h, \quad y_2=\frac{(h-1)^2}{h},\\
x_3=2h-1, \quad y_3=2(h-1),\quad
x_4=\frac{2h-1}{h}, \quad y_4=0.
\end{gather*}
We observe that the singular points $M_{1,2}$ are located on the hyperbola,
$M_1$ is located on the lines $\mathcal{L}_1=0$ and $\mathcal{L}_3=0$,
$M_2$ is located on the line $\mathcal{L}_2=0$, $M_3$ is located on the
line $\mathcal{L}_1=0$ and $M_4$ is located on the lines $\mathcal{L}_2=0$
 and $\mathcal{L}_3=0$.

Concerning the position of these singularities in relation to the invariant
lines and the invariant hyperbola, we have:
\begin{itemize}
 \item the location of $M_1$ and $M_2$ on the branches of the hyperbola:
$\operatorname{sign}(x_1x_2)=\operatorname{sign}(h)$;

 \item $M_3$ and $M_4$ could not belong to the hyperbola,
since $\Phi(x_3,y_3)=2(1-h)\ne0$ and $\Phi(x_4,y_4)=(h-1)^2/h^2\ne0$,
because of \eqref{cond:CSa-3-3IL};

 \item the position of the line $\mathcal{L}_2=0$ with respect to the line
$\mathcal{L}_1=0$: $\operatorname{sign}(\mathcal{L}_1-\mathcal{L}_2)
=\operatorname{sign}\big(h(h-1)\big)$;

 \item the position of $M_1$ and $M_4$ on $\mathcal{L}_3=0$:
$\operatorname{sign}(x_1-x_4)=\operatorname{sign}\big(h(1-h)\big)$;

 \item the position of $M_2$ and $M_4$ on $\mathcal{L}_2=0$:
$\operatorname{sign}(x_2-x_4)=\operatorname{sign}(h)$;

 \item the position of $M_1$ and $M_3$ on $\mathcal{L}_1=0$:
$\operatorname{sign}(x_1-x_3)=\operatorname{sign}(1-h)$.
\end{itemize}

On the other hand, for systems \eqref{sys:CSa-3-3IL}, we calculate the
invariant polynomials
$$
\begin{gathered}
\chi_C^{(7)}=2160\,h (h-1)^6 (2h+1)^2,\quad
N=9(h-1)(h+1)(x-y)^2.
\end{gathered}
$$
We observe that the condition $\chi_C^{(7)}<0$ implies that
$\operatorname{sign}(h-1)$ is controlled and we have the unique configuration
given by Config. H.111.

In the case $\chi_C^{(7)}>0$, we obtain the configuration
Config. H.112 if $N<0$ and Config. H.110 if $N>0$.


($\beta2$) {\it Possibility $\mu_0=0$.}
Then $h=0$ and, since we also obtain $\mu_1=0$ and $\mu_2=xy$,
two finite singularities of systems \eqref{sys:CSa-3} have gone
to infinity and collapsed with $[1,0,0]$ and $[0,1,0]$.
The remaining two finite singularities have the coordinates $M_i(x_i,y_i)$:
$$
x_1=-1, \quad y_1=-2,\quad
x_2=1, \quad y_2=0.
$$
In this case, the invariant hyperbola remains the same (since it
does not depend on $h$), whereas the invariant line
$\mathcal{L}_2=0$ goes to infinity and hence the line of infinity
$Z=0$ becomes double and we obtain only one configuration given by
Config. H.116.


(a2) {\it Case $\delta_4=0$.} In this case, the condition
$(h-1)^2(2h+1)+a(3h+1)^2=0$ yields $a=-(h-1)^2(2h+1))/(3h+1)^2$,
 which leads to the family of systems
\begin{equation} \label{sys:CSa-3-delta4=0}
\begin{gathered}
 \frac {dx}{dt}= \frac{2(h+1)^3(1-2h)}{(3h+1)^2}+(3h-1)x-hx^2+(h-1)xy,\\
 \frac {dy}{dt}= \frac{2(1-h)^3(2h+1)}{(3h+1)^2}+(3h-1)y-(h+1)xy+hy^2,
\end{gathered}
\end{equation}
with the condition
\begin{equation} \label{cond:CSa-3-delta4=0}
(h-1)(h+1)(2h-1)(2h+1)(3h-1)(3h+1)\ne0.
\end{equation}
These systems possess two invariant lines and two invariant hyperbolas
\begin{equation} \label{Hyp:3a-delta4=0}
\begin{gathered}
\mathcal{L}_1(x,y)=x-y-\frac{4h}{3h+1}=0,\quad
\mathcal{L}_2(x,y)=x-y-\frac{5h^2-1}{h(3h+1)}=0,\\
\Phi_1(x,y)=\frac{2(h+1)^3}{(3h+1)^2}-2x+x(x-y)=0,\\
\Phi_2(x,y)=\frac{2(1-h)^3}{(3h+1)^2}+\frac{2(3h-1)}{3h+1}\,x-y(x-y)=0.
\end{gathered}
\end{equation}

Since $\mu_0=h^2$, we consider again the possibilities:
$\mu_0\ne0$ and $\mu_0=0$.


($\alpha1$) {\it Subcase $\mu_0\ne0$.} Then $h\ne0$ and the four finite
singularities of systems \eqref{sys:CSa-3-delta4=0} have
coordinates $M_i(x_i,y_i)$, where:
\begin{gather*}
x_1=\frac{(h+1)^2}{3h+1},\quad
y_1=\frac{(h-1)^2}{3h+1},\quad
x_2=\frac{2h(h+1)}{3h+1},\quad
y_2=\frac{(2h+1)(h-1)^2}{h(3h+1)},\\
x_3=\frac{2(h+1)(2h-1)}{3h+1},\quad
y_3=\frac{2(h-1)(2h+1)}{3h+1},\\
x_4=\frac{(2h-1)(h+1)^2}{h(3h+1)},\quad
y_4=\frac{2h(h-1)}{3h+1}.
\end{gather*}

We observe that the singular point $M_1$ is located on both
hyperbolas and on the line $\mathcal{L}_1=0$, $M_2$ is located on
the hyperbola $\Phi_1=0$ and on the line $\mathcal{L}_2=0$, $M_3$
is located on the line $\mathcal{L}_1=0$ and $M_4$ is located on
the hyperbola $\Phi_2=0$ and on the line $\mathcal{L}_2=0$.

Concerning the position of the singular points on the lines and
hyperbolas, we observe that the position of $M_1$ and $M_3$ on
$\mathcal{L}_1=0$ is governed by
$\operatorname{sign}(x_1-x_3)=\operatorname{sign}\big((h-1)(h+1)(3h+1)\big)$
and the position
of $M_2$ and $M_4$ on $\mathcal{L}_2=0$ is governed by
$\operatorname{sign}(x_2-x_4)=\operatorname{sign}\big(h(h-1)(h+1)(3h+1)\big)$. 
Moreover, the
position of the hyperbolas is governed by
$\operatorname{sign}\big((h-1)(h+1)\big)$.

We observe that, in the case $(h-1)(h+1)<0$, we have $-1<h<1$.
Then, analyzing the sign of the expression $h(3h+1)$, we verify
that all the possible configurations for these values of the
parameter coincide. Analogously, we obtain the same configurations
by analyzing the sign of $h(3h+1)$ subjected to $(h-1)(h+1)>0$.
So, it is sufficient to only study $\operatorname{sign}\big((h-1)(h+1)\big)$.

Thus, we conclude that $\operatorname{sign}(N)
=\operatorname{sign}\big((h-1)(h+1)\big)$ and we
arrive at the configuration given by Config. H.140 if
$N<0$ and Config. H.139 if $N>0$.


($\beta$) {\it Subcase $\mu_0=0$.} Then $h=0$ and
two finite singular points have gone to infinity and coalesced
with $[1,0,0]$ and $[0,1,0]$, since $\mu_1=0$ and $\mu_2=xy$. The
remaining two finite singularities have the coordinates
$M_i(x_i,y_i)$, where
$$
x_1=-2, \quad y_1=-2,\quad
x_2=1, \quad y_2=1.
$$

In this case, both invariant hyperbolas remain the same (since
they do not depend on $h$), whereas the invariant line
$\mathcal{L}_2=0$ goes to infinity and hence the line of infinity
$Z=0$ becomes double (see Lemma \ref{lem:line-L}) and we obtain
only one configuration given by Config. H.146.


(b) {\it Possibility $\beta_7=0$.} We recall that
the conditions $\beta_6=0$ and $\beta_2\ne0$ imply $f=c \ne0$,
and then we arrive at systems
\eqref{sys:eta-poz-theta=0-gamma1=0-beta2ne0}. As earlier, via a
time rescaling, we may assume $c=1$. Moreover, the condition
$\beta_7=0$ implies $(2h-1)(2h+1)=0$ and, without loss of
generality, we could choose $h=1/2$, otherwise we apply the change
$(x,y,t,a,b,h)\mapsto(-y,-x,-t,b,a,-h)$, which keeps the systems
\eqref{sys:eta-poz-theta=0-gamma1=0-beta2ne0} and changes the sign of
$h$.

Now, according to Theorem \ref{trm:exist-hyp}, for the existence
of at least one hyperbola for systems
\eqref{sys:eta-poz-theta=0-gamma1=0-beta2ne0}, it is necessary and
sufficient the conditions $\gamma_9=0$ and $\mathcal{R}_8\ne0$.
So, we calculate $\ \gamma_9=3a/2 $\ and, setting $a=0$, we obtain
the 1-parameter family of systems
\begin{equation} \label{sys:CSa-4}
\frac {dx}{dt}= x-x^2/2-xy/2,\quad
 \frac {dy}{dt}= b+y-3xy/2+y^2/2,
\end{equation}
with the condition $b+4\ne0$.
These systems possess three invariant lines (two of them being parallel)
and an invariant hyperbola
\begin{equation} \label{Hyp:4a2}
\begin{gathered}
\mathcal{L}_{1,2}(x,y)=(x-y)^2-2(x-y)+2b=0,\quad
\mathcal{L}_3(x,y)=x=0,\\
\Phi(x,y)=4+b-4x+x(x-y)=0.
\end{gathered}
\end{equation}

We remark that, since
$\operatorname{Discriminant}\big[\mathcal{L}_{1,2}(x,y),x-y\big]=4(1-2b)$,
these lines are complex (respectively real) if $2b-1<0$
(respectively $2b-1>0$).

We calculate $\ \delta_5=3(8-25b)/2 $ and we consider two cases:
$\delta_5\ne0$ and $\delta_5=0$.

(b1) {\it Case $\delta_5\ne0$.} In this case we have $25b-8\ne0$
and hence $\Phi(x,y)=0$ (see \eqref{Hyp:4a}) is the unique invariant hyperbola.
Since $B_1=B_2=0$ for systems \eqref{sys:CSa-4}, we calculate
$$
B_3=-27\,b\,x^2 (x-y)^2/4.
$$

($\alpha$) {\it Subcase $B_3\ne0$.} Then $b\ne0$
and, since $\mu_0=1/4$, the finite singularities $M_i(x_i,y_i)$
of systems \eqref{sys:CSa-4} are of total multiplicity 4, and
their coordinates are
$$
x_{1,2}=\frac{3\pm\sqrt{1-2b}}{2}, \quad
y_{1,2}=\frac{1\mp\sqrt{1-2b}}{2},\quad
x_{3,4}=0, \quad
y_{3,4}=-1\pm\sqrt{1-2b}.
$$

We observe that the singular points $M_{1,2}$ are located on the
hyperbola and on the invariant lines $\mathcal{L}_{1,2}=0$,
whereas the singularities $M_{3,4}$ are located on the
intersections of the couple of parallel invariant lines with the
third one.

Considering the singular points $M_{1,2}$, we see that
$x_1x_2=(b+4)/2$ and hence $\operatorname{sign}(b+4)$ detects the location of
these singularities on the same or different branches of the
hyperbola. Moreover, the position of the hyperbola is governed by
$\operatorname{sign}(b+4)$.

To detect when the singularities $M_{3,4}$ also belong to
the hyperbola, we consider \eqref{Hyp:4a} and we calculate
$$
\big[\Phi(x_3,y_3)\big]\big[\Phi(x_4,y_4)\big]=(b+4)^2\ne0,
$$
otherwise the hyperbola splits into two lines. Thus none of the
singular points $M_3$ or $M_4$ could belong to the hyperbola
\eqref{Hyp:4a}.

On the other hand, for systems \eqref{sys:CSa-4}, we calculate the
invariant polynomials
$$
\chi_A^{(7)}=9(1-2b)/256,\quad
\chi_C^{(7)}=135(b+4)(25b-8)^2/8
$$
and we conclude that $\operatorname{sign}(\chi_A^{(7)})
=\operatorname{sign}(1-2b)$ (if
$2b-1\ne0$) and from $\delta_5\ne0$ (i.e. $25b-8\ne0$) we have
$\operatorname{sign}(\chi_C^{(7)})=\operatorname{sign}(b+4)$.


($\alpha1$) {\it Possibility $\chi_A^{(7)}<0$.}
Then all four finite singularities are complex as well as the
invariant lines $\mathcal{L}_{1,2}=0$ and we get the configuration
shown in Config. H.115.


($\alpha2$) {\it Possibility $\chi_A^{(7)}>0$.}
Then all four finite singularities and the invariant lines
$\mathcal{L}_{1,2}=0$ are real and we obtain the configuration
Config. H.114 if $\chi_C^{(7)}<0$ and
Config. H.113 if $\chi_C^{(7)}>0$.


($\alpha3$) {\it Possibility $\chi_A^{(7)}=0$.}
Then we have two double finite singular points (namely,
$M_1$=$M_2$ and $M_3$=$M_4$) and also the invariant lines
$\mathcal{L}_{1,2}=0$ coalesce and we obtain a double invariant
line. So, we arrive at the configuration Config. H.117.


($\beta$) {\it Subcase $B_3=0$.} Then $b=0$ and we
obtain a specific system possessing a fourth invariant line,
namely $\mathcal{L}_4=y=0$. Then, we obtain the unique
configuration Config. H.119.


(b2) {\it Case $\delta_5=0$.} Then $b=8/25$ and
again we obtain a concrete system, but now possessing a second
hyperbola, namely $\Phi_2(x,y)=-4/25-4y/5+y(x-y)=0$. Moreover, we
observe that, for systems \eqref{sys:CSa-4} with $b=8/25$, we have
$B_3=-54x^2(x-y)^2/25\ne0$ and hence there are no more invariant
lines rather than the ones given in \eqref{Hyp:4a}. So, we arrive
at the unique configuration Config. H.147.

\subparagraph{Subcase $\beta_2=0$.} Then $c=0$ and we obtain the systems
\begin{equation} \label{sys:eta-poz-theta=0-gamma1=0-beta2=0}
\frac {dx}{dt}= a-hx^2+(h-1)xy,\quad
 \frac {dy}{dt}= b-(h+1)xy+hy^2.
\end{equation}


(a) {\it Possibility $\beta_7\ne0$.} Then $(2h-1)(2h+1)\ne0$ and,
since $\beta_{10}=-2(3h-1)(3h+1)$, we consider two cases: $\beta_{10}\ne0$
and $\beta_{10}=0$.


(a1) {\it Case $\beta_{10}\ne0$.} Then $(3h-1)(3h+1)\ne0$ and, according
to Theorem \ref{trm:exist-hyp}, for the existence of at least one
invariant hyperbola for systems \eqref{sys:eta-poz-theta=0-gamma1=0-beta2=0},
it is necessary and sufficient the conditions $\gamma_7\gamma_8=0$ and
 $\mathcal{R}_5\ne0$. So, we calculate
\begin{gather*}
\gamma_7=8(h-1)(h+1)\,\mathcal{E}_1,\quad
\gamma_8=42(h-1)(h+1)(3h-1)^2(3h+1)^2\,\mathcal{E}_2\mathcal{E}_3,\\
\mathcal{E}_1=a(2h+1)+b(2h-1),\quad \mathcal{E}_2=2a(1-h)+b(2h-1),\\
 \mathcal{E}_3=2b(h+1)-a(2h+1).
\end{gather*}
We observe that we can pass from the condition $\mathcal{E}_2=0$
to the condition $\mathcal{E}_3=0$ via the change $(x,y,a,b,h)\mapsto(y,x,b,a,-h)$,
and any of these conditions is equivalent to $\gamma_8=0$.
However, the condition $\mathcal{E}_1=0$ could not be replaced.
So, we need to analyze the possibility $\gamma_7=0$ and then the possibility
 $\gamma_8=0$.
We calculate
$$
\beta_8=-6(4h-1)(4h+1),\quad
\delta_2=2\big[(a + b) (128 h^2-11 ) + (a - b) h (400 h^2-49)\big] .
$$


($\alpha$) {\it Subcase $\beta_8^2+\delta_2^2\ne0$.}
By Theorem \ref{trm:exist-hyp} (see  Diagram \ref{diagr:eta-poz} in
this case systems \eqref{sys:eta-poz-theta=0-gamma1=0-beta2=0}
possess a single invariant hyperbola if and only if
$\gamma_7\gamma_8=0$ and $\mathcal{R}_5\ne0$. We consider the
cases $\gamma_7=0$ and $\gamma_8=0$ separately.


($\alpha1$) {\it Possibility $\gamma_7=0$.} Then $\mathcal{E}_1=0$ and
we obtain a subfamily of systems \eqref{sys:CSa-1} with $c=0$.
So, we arrive at the 2-parameter family of systems
\begin{equation} \label{sys:CSa-5}
\begin{gathered}
 \frac {dx}{dt}= a(2h-1)-hx^2+(h-1)xy,\\
 \frac {dy}{dt}= -a(2h+1)-(h+1)xy+hy^2,
\end{gathered}
\end{equation}
for which $h\ne0$, otherwise we get degenerate systems, and considering
the condition $N\beta_7\beta_{10}\mathcal{R}_5(\beta_8^2+\delta_2^2)\ne0$, we have
\begin{equation} \label{cond:CSa-5}
ah(h-1)(h+1)(2h-1)(2h+1)(3h-1)(3h+1)(4h-1)(4h+1)\ne0.
\end{equation}
These systems possess two parallel invariant lines and the invariant hyperbola
\begin{equation} \label{Hyp:5a}
\mathcal{L}_{1,2}=(x-y)^2-4a=0,\quad \Phi(x,y)=a+xy=0.
\end{equation}
Since $\mu_0=h^2\ne0$, these systems possess all four finite
singularities on the finite part of the phase plane and their
coordinates are $M_i(x_i,y_i)$, where
$$
x_{1,2}=\pm\sqrt{a}, \quad
y_{1,2}=\mp\sqrt{a},\quad
x_{3,4}=\pm(2h-1)\sqrt{a}, \quad
y_{3,4}=\pm(2h+1)\sqrt{a}.
$$

We observe that the singular points $M_{1,2}$ are located on the
hyperbola and on the invariant lines $\mathcal{L}_{1,2}=0$,
whereas the singularities $M_{3,4}$ are located only on the
invariant lines.

Considering the singular points $M_{1,2}$, we see that $x_1x_2=-a$
and hence $\operatorname{sign}(a)$ detects the location of these singularities
on the same or different branches of the hyperbola. Moreover, the
position of the hyperbola is also governed by $\operatorname{sign}(a)$.

We point out that the singularities $M_{3,4}$ could not belong to
the hyperbola since
$$
\big[\Phi(x_3,y_3)\big]\big[\Phi(x_4,y_4)\big]=16a^2h^4\ne0,
$$
because of conditions \eqref{cond:CSa-5}. On the other hand, we
calculate $\ \chi_A^{(2)}=80ah^6 $\ and we note that
$\operatorname{sign}(\chi_A^{(2)})=\operatorname{sign}(a)$. So, we arrive at the
configurations given by Config. H.80 if $\chi_A^{(2)}<0$
and Config. H.91 if $\chi_A^{(2)}>0$.


($\alpha2$) {\it Possibility $\gamma_8=0$.} Then
$\mathcal{E}_2=0$ and this is equivalent to the relations
$a=a_1(2h-1)$ and $b=2a_1(h-1)$, where $a_1$ is a new parameter.
So, setting this reparametrization in
\eqref{sys:eta-poz-theta=0-gamma1=0-beta2=0} and replacing the old
parameter $a$ instead of $a_1$, we arrive at the 2-parameter
family of systems
\begin{equation} \label{sys:CSa-6}
\begin{aligned}
 \frac {dx}{dt}= a(2h-1)-hx^2+(h-1)xy,\\
 \frac {dy}{dt}= 2a(h-1)-(h+1)xy+hy^2,
\end{aligned}
\end{equation}
with the conditions
\begin{equation} \label{cond:CSa-6}
a(h-1)(h+1)(2h-1)(2h+1)(3h-1)(3h+1)(4h-1)(4h+1)\ne0.
\end{equation}
These systems possess two parallel invariant lines and the invariant hyperbola
\begin{equation} \label{Hyp:6a}
\mathcal{L}_{1,2}=(x-y)^2-a/h=0,\quad \Phi(x,y)=a-x(x-y)=0.
\end{equation}

We consider the coordinates $M_i(x_i,y_i)$ of the finite singular
points of systems \eqref{sys:CSa-6}:
\begin{gather*}
x_{1,2}=\pm\sqrt{ah}, \quad
y_{1,2}=\pm\frac{(h-1)\sqrt{ah}}{h},\\
x_{3,4}=\pm\frac{(2h-1)\sqrt{ah}}{h}, \quad
y_{3,4}=\pm2\sqrt{ah}.
\end{gather*}

We observe that the singular points $M_{1,2}$ are located on the
hyperbola and on the invariant lines $\mathcal{L}_{1,2}=0$,
whereas the singularities $M_{3,4}$ are located only on the
invariant lines.

Considering the singular points $M_{1,2}$, we see that
$x_1x_2=-ah$ and hence $\operatorname{sign}(ah)$ detects the location of these
singularities on the same or different branches of the hyperbola.
Moreover, the position of the hyperbola is governed by $\operatorname{sign}(a)$.

We remark that the singular points $M_{3,4}$ could not belong to
the hyperbola since
$$
\big[\Phi(x_3,y_3)\big]\big[\Phi(x_4,y_4)\big]=\frac{a^2(3h-1)^2}{h^2}\ne0,
$$
because of conditions \eqref{cond:CSa-6}. On the other hand, we
calculate
$$
\chi_A^{(7)}=ah(h-1)^2(h+1)^2/4,\quad
\chi_B^{(7)}=6480\,a^3 (h-1)^2 (3h+1)^4
$$
and we note that $\operatorname{sign}(\chi_A^{(7)})=\operatorname{sign}(ah)$
and $\operatorname{sign}(\chi_B^{(7)})=\operatorname{sign}(a)$.

If $\chi_A^{(7)}\ne0$ (i.e. $h\ne0$), we obtain the following conditions
and configurations:
\begin{itemize}
 \item $\chi_A^{(7)}<0$ and $\chi_B^{(7)}<0$ $\Rightarrow$  Config. H.78;
 \item $\chi_A^{(7)}<0$ and $\chi_B^{(7)}>0$ $\Rightarrow$  Config. H.79;
 \item $\chi_A^{(7)}>0$ and $\chi_B^{(7)}<0$ $\Rightarrow$  Config. H.96;
 \item $\chi_A^{(7)}>0$ and $\chi_B^{(7)}>0$ $\Rightarrow$  Config. H.95.
\end{itemize}

In the case $\chi_A^{(7)}=0$ (i.e. $h=0$), then we have
$\mu_0=\mu_1=\mu_2=\mu_3=0$ and $\mu_4=a^2x^2y^2\ne0$.
Thus, the four finite singularities have gone to infinity and two of them
coalesced with $[1,0,0]$ and the other two of them coalesced with $[0,1,0]$.
 Moreover, the two invariant lines $\mathcal{L}_{1,2}=0$ have also gone
to infinity and hence the line of infinity $Z=0$ is a triple invariant
line for the system, because $Z^2$ is a double factor of the polynomials
$\mathcal{E}_1$ and $\mathcal{E}_2$ (see Lemma \ref{lem:line-L}).

Now, according to the $\operatorname{sign}(a)$ we have different position of the
hyperbola and consequently distinct configurations. So, we get the
configurations shown by Config. H.108 if
$\chi_B^{(7)}<0$ and by Config. H.109 if
$\chi_B^{(7)}>0$.

($\beta$) {\it Subcase $\beta_8=\delta_2=0$.} Then
the condition $\beta_8=0$ gives $(4h-1)(4h+1)=0$ and, without loss
of generality, we may assume $h=1/4$ by the change
$(x,y,a,b,h)\mapsto(y,x,b,a,-h)$.

We calculate
$$
\delta_2= 6(b-3a),\quad \gamma_7=-15(3a-b)/4,\quad
\gamma_8=15435(3a-5b)(3a-b)/8192
$$
and hence the condition $\delta_2=0$ yields $b=3a$ and then
$\gamma_7=\gamma_8=0$. So we obtain the 1-parameter family of
systems
\begin{equation} \label{sys:CSa-7}
\frac {dx}{dt}= a-x^2/4-3xy/4,\quad
 \frac {dy}{dt}= 3a-5xy/4+y^2/4,
\end{equation}
with the condition $a\ne0$.
These systems possess two parallel invariant lines and two invariant hyperbolas
\begin{equation} \label{Hyp:7a}
\begin{gathered}
\mathcal{L}_{1,2}=(x-y)^2+8a=0,\quad
\Phi_1(x,y)=2a-xy=0,\\
\Phi_2(x,y)=2a+x(x-y)=0.
\end{gathered}
\end{equation}

Since $\mu_0=1/16\ne0$, all the four finite singularities are on the finite
part of the phase plane and their coordinates are $M_i(x_i,y_i)$:
$$
x_{1,2}=\pm\sqrt{-2a}, \quad
y_{1,2}=\mp\sqrt{-2a},\quad
x_{3,4}=\pm\frac{\sqrt{-2a}}{2}, \quad
y_{3,4}=\mp\frac{3\sqrt{-2a}}{2}.
$$

We observe that the singular points $M_{1,2}$ are located on the
first hyperbola $\Phi_1=0$, whereas the singularities $M_{3,4}$
are located on the second hyperbola $\Phi_2=0$. All singular
points are located on the invariant lines $\mathcal{L}_{1,2}=0$.

Considering the singular points $M_{1,2}$ (respectively
$M_{3,4}$), we see that $x_1x_2=2a$ (respectively $x_3x_4=a/2$)
and hence $\operatorname{sign}(a)$ detects the location of these singularities
to be on the same or different branches of the hyperbolas that
they are located on. Moreover, the position of the hyperbola is
also governed by $\operatorname{sign}(a)$.

We remark that the singular points $M_{1,2}$ (respectively
$M_{3,4}$) could not belong to the hyperbola $\Phi_2=0$
(respectively $\Phi_1=0$) since
$$
\big[\Phi_2(x_1,y_1)\big]\big[\Phi_2(x_2,y_2)\big]=4a^2\ne0,\quad
\big[\Phi_1(x_3,y_3)\big]\big[\Phi_1(x_4,y_4)\big]=a^2/4\ne0,
$$
because $a\ne0$.

On the other hand, we calculate
\[
\chi_A^{(7)}=-225a/2048
\]
and we note that $\operatorname{sign}(\chi_A^{(7)})=-\operatorname{sign}(a)$. 
So, we get the configurations shown by Config. H.143 if
$\chi_A^{(7)}<0$ and Config. H.141 if $\chi_A^{(7)}>0$.


(a2) {\it Case $\beta_{10}=0$.} Then
$(3h-1)(3h+1)=0$ and, without loss of generality, we may assume
$h=1/3$, since the case $h=-1/3$ could be brought to the case
$h=1/3$ via the change $(x,y,a,b,h)\mapsto(y,x,b,a,-h)$. So, we
arrive at the systems
\begin{equation} \label{sys:CSa-8}
\frac {dx}{dt}= a-x^2/3-2xy/3,\quad
 \frac {dy}{dt}= b-4xy/3+y^2/3.
\end{equation}
with the condition $a\ne0$, possessing a pair of parallel
invariant lines and a couple of invariant hyperbolas with parallel
asymptotes
\begin{equation} \label{Hyp:8a}
\begin{gathered}
\mathcal{L}_{1,2}(x,y)=(x-y)^2-3(a-b)=0,\\
\Phi_{1,2}(x,y)=3a\pm\sqrt{3(4a-b)}x+x(x-y)=0.
\end{gathered}
\end{equation}

In accordance to Theorem \ref{trm:exist-hyp}, we have to analyze
the following subcases: $\gamma_7\ne0$ and $\gamma_7=0$ and we
calculate
$$
\gamma_7=-65(5a-b)/27,\quad \gamma_{10}=8(4a-b)/27.
$$


($\alpha$) {\it Subcase $\gamma_7\ne0$.} Then we
could not have other invariant hyperbolas rather than the ones in
\eqref{Hyp:8a}. Moreover, the hyperbolas \eqref{Hyp:8a} are
complex if $\gamma_{10}<0$, real if $\gamma_{10}>0$ and they
coincide if $\gamma_{10}=0$. Then, we consider two possibilities:
$\gamma_{10}<0$ and $\gamma_{10}\ge0$.


($\alpha1$) {\it Possibility $\gamma_{10}<0$.}
 Then the hyperbolas \eqref{Hyp:8a} are complex. In this case, we
set a new parameter $v\ne0$ satisfying $4a-b=-3v^2$, which yields
$b=4a+3v^2$ and we obtain the 2-parameter family of systems
\begin{equation} \label{sys:CSa-8A}
\frac {dx}{dt}= a-x^2/3-2xy/3,\quad
 \frac {dy}{dt}= 4a+3v^2-4xy/3+y^2/3,
\end{equation}
with the condition $av\ne0$, possessing the invariant lines and
invariant hyperbolas
\begin{equation} \label{Hyp:8a-A}
\begin{gathered}
\mathcal{L}_{1,2}(x,y)=(x-y)^2+9(a+v^2)=0,\\
\Phi_{1,2}(x,y)=3a\pm3ivx+x(x-y)=0.
\end{gathered}
\end{equation}

We calculate
$$
\mu_0=1/9,\quad B_1=0,\quad
B_2=-512\,a(4a+3v^2)(x-y)^4
$$
and we consider two cases: $B_2\ne0$ and $B_2=0$.


(i) {\it Case $B_2\ne0$.} Then there are no other invariant lines
 rather than $\mathcal{L}_{1,2}=0$ in \eqref{Hyp:8a-A}. We calculate
\begin{gather*}
\mu_0=1/9\ne0,\quad
{\rm D}=-4096\,v^4(a+v^2)^2/3,\\
{\rm S}=256\,v^2(a+v^2)(x-y)^2(2x+y)^2/2187,\\
{\rm R}=-16\big[(4a+5v^2)x^2+2(2a+v^2)xy+(a+2v^2)y^2\big]/81,\\
{\rm T}=-81{\rm R}{\rm S}/32.
\end{gather*}
We claim that all four finite singular points are complex.
Indeed, if $a+v^2>0$, we observe that
\begin{gather*}
\operatorname{Discriminant}\big[{\rm R},x\big]=-1024v^2(a+v^2)y^2/729<0,\\
\operatorname{Coefficient}\big[{\rm R},y^2\big]=-16(a+2v^2)/81<0
\end{gather*}
and hence ${\rm R}<0$. Since ${\rm D}<0$, by Proposition
\ref{prop:number-FSPs} all four finite singularities of systems
\eqref{sys:CSa-8A} are complex.

Now, if $a+v^2<0$, then ${\rm D}<0$ and ${\rm S}<0$, and by Proposition
\ref{prop:number-FSPs} all four finite singularities of systems
\eqref{sys:CSa-8A} are complex.

Finally, if $a+v^2=0$, then ${\rm D}={\rm T}=0$ and we have two collisions
of finite singular points, i.e. we have two double singular
points. As in any case we have only complex singularities, these
double singular points are also complex. So, our claim is proved.

We calculate $\chi_A^{(7)}=-16(a+v^2)/81 $ and we note that
$\operatorname{sign}(\chi_A^{(7)})=-\operatorname{sign}(a+v^2)$.

If $\chi_A^{(7)}<0$, then the invariant lines are also complex and
we get the configuration Config. H.144. In the case
$\chi_A^{(7)}>0$ the invariant lines are real and we arrive at
the configuration Config. H.145. If $\chi_A^{(7)}=0$,
then the invariant lines coalesce and become a double line, which leads
to configuration Config. H.153.


(ii) {\it Case $B_2=0$.} Then $4a+3v^2=0$ and
systems \eqref{sys:CSa-8A} have a third invariant line $y=0$ and
the lines $\mathcal{L}_{1,2}=0$ are complex. So, we get the
configuration Config. H.151.


($\alpha2$) {\it Possibility $\gamma_{10}>0$.}
In this case, we set the new parameter $v\ne0$ satisfying $4a-b=3v^2$,
which yields $b=4a-3v^2$ and we obtain the 2-parameter family of systems
\begin{equation} \label{sys:CSa-8B}
\frac {dx}{dt}= a-x^2/3-2xy/3,\quad
 \frac {dy}{dt}= 4a-3v^2-4xy/3+y^2/3,
\end{equation}
with the condition $a\ne0$, possessing the invariant lines and
invariant hyperbolas
\begin{equation} \label{Hyp:8a-B}
\begin{gathered}
\mathcal{L}_{1,2}(x,y)=(x-y)^2+9(a-v^2)=0,\\
\Phi_{1,2}(x,y)=3a\pm3vx+x(x-y)=0.
\end{gathered}
\end{equation}

\begin{remark} \label{rem:CSa-8B-v=0} \rm
We remark that, the condition $v=0$ for systems
\eqref{sys:CSa-8B} is equivalent to $\gamma_{10}=0$.
\end{remark}

We calculate
$$
\mu_0=1/9,\quad B_1=0,\quad B_2=-512 a(4a-3v^2)(x-y)^4
$$
and we consider two cases: $B_2\ne0$ and $B_2=0$.


(i) {\it Case $B_2\ne0$.} Then there is no other
invariant line rather than $\mathcal{L}_{1,2}=0$ in
\eqref{Hyp:8a-B}. Since $\mu_0\ne0$, all four finite singularities
of systems \eqref{sys:CSa-8B} are on the finite part of the phase
plane and their coordinates are $M_i(x_i,y_i)$, where
\begin{gather*}
x_{1,2}=-v\pm\sqrt{v^2-a}, \quad
y_{1,2}=-v\mp2\sqrt{v^2-a},\\
x_{3,4}=v\pm\sqrt{v^2-a}, \quad
y_{3,4}=v\mp2\sqrt{v^2-a}.
\end{gather*}

We observe that the singular points $M_{1,2}$ are located on the
first hyperbola $\Phi_1=0$ and on the invariant lines
$\mathcal{L}_{1,2}=0$, whereas the singularities $M_{3,4}$ are
located on the second hyperbola $\Phi_2=0$ and on the invariant
lines $\mathcal{L}_{1,2}=0$.

Considering the pairs of singular points $M_{1,2}$ and $M_{3,4}$,
we see that $x_1x_2=x_3x_4=a$ and hence $\operatorname{sign}(a)$ detects the
location of these singularities to be on the same or different
branches of the respective hyperbola they are located on.

We remark that the singular points $M_{1,2}$ (respectively $M_{3,4}$)
could belong to the hyperbola $\Phi_2=0$ (respectively $\Phi_1=0$)
if and only if
\begin{gather*}
\big[\Phi_2(x_1,y_1)\big]\big[\Phi_2(x_2,y_2)\big]=36av^2=0,\\
\big[\Phi_1(x_3,y_3)\big]\big[\Phi_1(x_4,y_4)\big]=36av^2=0,
\end{gather*}
which are equivalent to $v=0$. However by Remark
\ref{rem:CSa-8B-v=0} and the condition $\gamma_{10}>0$ we have
$v\ne0$.

On the other hand, we calculate
$$
\chi_A^{(7)}=16(v^2-a)/81,\quad
\chi_C^{(3)}=17039360\,a(a+3v^2)^2/9
$$
and we conclude that $\operatorname{sign}(\chi_A^{(7)})
=\operatorname{sign}(v^2-a)$ and
 $\operatorname{sign}(\chi_C^{(3)})=\operatorname{sign}(a)$.

Since $v\ne0$, the invariant hyperbolas $\Phi_{1,2}=0$ are
distinct. We observe that the condition $\chi_A^{(7)}\le0$
implies $a>0$ (as $v\ne0$) and consequently, $\chi_C^{(3)}>0$.
Moreover, if $\chi_A^{(7)}=0$, then both invariant lines coalesce
and we obtain the double invariant line $(x-y)^2=0$. So, we arrive
at the following conditions and configurations:
\begin{itemize}
 \item $\chi_A^{(7)}<0$ $\Rightarrow$  Config. H.142;
 \item $\chi_A^{(7)}>0$ and $\chi_C^{(3)}<0$ $\Rightarrow$  Config. H.137;
 \item $\chi_A^{(7)}>0$ and $\chi_C^{(3)}>0$ $\Rightarrow$  Config. H.138;
 \item $\chi_A^{(7)}=0$ $\Rightarrow$  Config. H.152.
\end{itemize}

(ii) {\it Case $B_2=0$.} Then $a=3v^2/4$ and we
have a third invariant line $\mathcal{L}_3(x,y)=y=0$ and the
previous two lines could be factored as
$\mathcal{L}_1(x,y)=2x-2y+3v=0$ and
$\mathcal{L}_2(x,y)=2x-2y-3v=0$.

Since $a>0$, we have
$$
\chi_A^{(7)}=4v^2/81>0,\quad \chi_C^{(3)}=19968000v^2>0
$$
and we obtain the unique configuration Config. H.149.


($\alpha3$) {\it Possibility $\gamma_{10}=0$.} In
this case according to Remark \ref{rem:CSa-8B-v=0} we have
$v=0$, and then $\chi_A^{(7)}=-16a/81\ne0$. In this case, the two
hyperbolas coalesce and we get a double hyperbola. Furthermore,
the singularities coalesce two by two and we have two double
singular points (namely $M_1=M_3$ and $M_2=M_4$).

It remains to observe that the condition $\chi_A^{(7)}<0$
(respectively $\chi_A^{(7)}>0$) implies $\chi_C^{(3)}>0$
(respectively $\chi_C^{(3)}<0$). So, we get the configuration
Config. H.155 if $\chi_A^{(7)}<0$ and
Config. H.154 if $\chi_A^{(7)}>0$.


($\beta$) {\it Subcase $\gamma_7=0$.} Then $b=5a$ and we arrive at
the 1-parameter family of systems
\begin{equation} \label{sys:CSa-9}
\frac {dx}{dt}= a-x^2/3-2xy/3,\quad
 \frac {dy}{dt}= 5a-4xy/3+y^2/3,
\end{equation}
with the condition $a\ne0$.

These systems possess a couple of parallel invariant lines, a pair
of invariant hyperbolas with parallel asymptotes presented in
\eqref{Hyp:8a} and a third hyperbola
\begin{equation} \label{Hyp:9a}
\begin{gathered}
\mathcal{L}_{1,2}(x,y)=(x-y)^2+12a=0,\\
\Phi_{1,2}(x,y)=3a\pm\sqrt{-3a}\,x+x(x-y)=0,\quad
\Phi_3(x,y)=xy-3a=0.
\end{gathered}
\end{equation}

Since $B_1=0$ and $B_2=-2560a^2(x-y)^4\ne0$, systems
\eqref{sys:CSa-9} could not possess other invariant lines
than the ones in \eqref{Hyp:9a}. Moreover, we have $\mu_0=1/9\ne0$
and all the four singularities are on the finite part of the phase
plane with coordinates $M_i(x_i,y_i)$, where
$$
x_{1,2}=\pm\sqrt{-3a}, \quad
y_{1,2}=\mp\sqrt{-3a},\quad
x_{3,4}=\pm\frac{\sqrt{-3a}}{3}, \quad
y_{3,4}=\mp\frac{5\sqrt{-3a}}{3}.
$$

We observe that all four singular points are located on the invariant
lines and also: $M_1$ is located on the hyperbolas $\Phi_2=0$ and
$\Phi_3=0$, $M_2$ is located on the hyperbolas $\Phi_1=0$ and $\Phi_3=0$,
$M_3$ is located on the hyperbola $\Phi_1=0$ and $M_4$ is located on the
hyperbola $\Phi_2=0$.

Concerning the position of the singularities on the hyperbolas, we have
\begin{itemize}
 \item the position of $M_2$ and $M_3$ on $\Phi_1(x,y)=0$ is controlled by
$\operatorname{sign}(x_2x_3)=\operatorname{sign}(a)$;
 \item the position of $M_1$ and $M_4$ on $\Phi_2(x,y)=0$ is controlled by
$\operatorname{sign}(x_1x_4)=\operatorname{sign}(a)$;
 \item the position of $M_1$ and $M_2$ on $\Phi_3(x,y)=0$ is controlled by
$\operatorname{sign}(x_2x_3)=\operatorname{sign}(3a)$.
\end{itemize}

We also point out that because $a\ne0$, the singularities could be
located on the hyperbolas only as it is described above.

We remark that, if $a>0$, then the four singularities are complex
as well as the pair of invariant hyperbolas $\Phi_{1,2}(x,y)=0$
and the couple of invariant lines $\mathcal{L}_{1,2}(x,y)=0$.

On the other hand, we calculate $\gamma_{10}=-8a/27 $ and we
conclude that $\operatorname{sign}(\gamma_{10})=-\operatorname{sign}(a)$. 
So, we arrive at the
configuration Config. H.159 if $\gamma_{10}<0$ and
Config. H.158 if $\gamma_{10}>0$.


(b) {\it Possibility $\beta_7=0$.} Then $(2h-1)(2h+1)=0$ and, without
loss of generality as earlier, we may assume $h=1/2$.
So, we obtain the systems
\begin{equation} \label{sys:eta-poz-theta=0-gamma1=0-beta2=0-beta7=0}
\frac {dx}{dt}= a-x^2/2-xy/2,\quad
 \frac {dy}{dt}= b-3xy/2+y^2/2.
\end{equation}

According to Theorem \ref{trm:exist-hyp}, the condition $\gamma_7=0$
is necessary and sufficient for the existence of invariant hyperbolas
for systems \eqref{sys:eta-poz-theta=0-gamma1=0-beta2=0-beta7=0}.
Moreover, this condition implies the existence of two such hyperbolas.

We calculate $\gamma_7=-12a=0 $ and we obtain the 1-parameter
family of systems
\begin{equation} \label{sys:CSa-10}
\frac {dx}{dt}= -x^2/2-xy/2,\quad
 \frac {dy}{dt}= b-3xy/2+y^2/2.
\end{equation}
with the condition $b\ne0$.

These systems possess three invariant lines and two invariant hyperbolas
\begin{equation} \label{Hyp:10a}
\begin{gathered}
\mathcal{L}_{1,2}(x,y)=(x-y)^2+2b=0,\quad
\mathcal{L}_3(x,y)=x=0,\\
\Phi_1(x,y)=b-2xy=0,\quad
\Phi_2(x,y)=b+x(x-y)=0.
\end{gathered}
\end{equation}

For systems
\eqref{sys:CSa-10} we calculate $B_1=B_2=0$ and $B_3=-27bx^2(x-y)^2/4\ne0$
and therefore by Lemma \ref{lem:Bi-0} systems
\eqref{sys:CSa-10} could not posses other invariant lines rather
than the ones in \eqref{Hyp:10a}. Since $\mu_0=1/4\ne0$, these
systems have finite singularities of total multiplicity 4 with
coordinates $M_i(x_i,y_i)$, where
$$
x_{1,2}=\pm\frac{\sqrt{-2b}}{2}, \quad
y_{1,2}=\mp\frac{\sqrt{-2b}}{2},\quad
x_{3,4}=0, \quad
y_{3,4}=\pm\sqrt{-2b}.
$$

We observe that the singular points $M_{1,2}$ are located on the
two hyperbolas and on the lines $\mathcal{L}_{1,2}=0$ and the
singularities $M_{3,4}$ are located on the three invariant lines.

Moreover, due $b\ne0$ we deduce that the singular points $M_{3,4}$
could not belong to the hyperbolas. By the same argument the
singular points $M_{1,2}$ could not belong to the invariant line
$\mathcal{L}_3=0$.

Since $x_1x_2=b/2$, the position of the singular points $M_{1,2}$
on the hyperbola is governed by $\operatorname{sign}(b)$, as well as the
position of the invariant hyperbolas.

We calculate $\ \chi_A^{(7)}=-9b/128 $\ and we conclude that
$\operatorname{sign}(\chi_A^{(7)})=\operatorname{sign}(b)$.

It is worth mentioning that, if $b>0$, then all four singular
points are complex as well as the couple of invariant lines
$\mathcal{L}_{1,2}=0$. So, we get the configuration
Config. H.150 if $\chi_A^{(7)}<0$ and
Config. H.148 if $\chi_A^{(7)}>0$.

\subsubsection{Possibility $N=0$}

Since for systems \eqref{sys:eta-poz-Gen} we have
$\theta=-(g-1)(h-1)(g+h)/2=0$, we observe that the condition
$$
N=(g-1)(g+1)x^2+2(g-1)(h-1)xy+(h-1)(h+1)y^2=0
$$
implies the vanishing of two factors of $\theta$. Then, without
loss of generality, we may assume $g=1=h$, otherwise in the case
$g+h=0$ and $g-1\ne0$ (respectively $h-1\ne0$), we apply the
change $(x,y,g,h)\mapsto(-y,x-y,1-g-h,g)$ (respectively
$(x,y,g,h)\mapsto(y-x,-x,h,1-g-h)$) which preserves the form of
such systems.

So, $g=h=1$ and by an additional translation we arrive at the systems
\begin{equation} \label{sys:eta-poz-N=0}
\frac {dx}{dt}= a+ dy+x^2,\quad
 \frac {dy}{dt}= b+ex +y^2,
\end{equation}
for which we calculate
$$
\beta_1=4de,\quad \beta_2=-2(d+e).
$$

According to Theorem \ref{trm:exist-hyp}, a necessary condition
for the existence of hyperbolas for these systems is $\beta_1=0$.
This condition is equivalent to $de=0$ and, without loss of
generality, we may assume $e=0$, by the change
$(x,y,a,b,d,e)\mapsto(y,x,b,a,e,d)$.

Then $\beta_2=-2d$ and we analyze two cases: $\beta_2\ne0$ and $\beta_2=0$.

\paragraph{Case $\beta_2\ne0$.} Then $d\ne0$ and via the rescaling
$(x,y,t)\mapsto(4d\,x,4d\,y,t/(4d))$, we may assume $d=4$. In this
case, since $\beta_1=0$, according to Theorem \ref{trm:exist-hyp}
the conditions $\gamma_{11}=0$ and $\mathcal{R}_9\ne0$ are
necessary and sufficient for the existence of one invariant
hyperbola.

We calculate $\gamma_{11}=-64(a-4b+1) $ and, setting $a=4b-1$,
we obtain the 1-parameter family of systems
\begin{equation} \label{sys:CSb-1}
\frac {dx}{dt}= 4b-1+4y+x^2,\quad  \frac {dy}{dt}= b+y^2,
\end{equation}
for which $\mathcal{R}_9=40 ( b+1)\ne0$. These systems possess the
invariant lines and the invariant hyperbola
\begin{equation} \label{Hyp:1b}
\mathcal{L}_{1,2}(x,y)=y^2+b=0,\quad
\Phi(x,y)=b-1-x+3y+y(x-y)=0.
\end{equation}

Since $B_1=0$ and $B_2=-124416(b+1)y^4\ne0$, systems
\eqref{sys:CSb-1} could not possess other invariant lines rather
than the ones in \eqref{Hyp:1b}. Moreover, $\mu_0=1\ne0$ implies
that these systems possess finite singularities $M_i(x_i,y_i)$
of total multiplicity four and their coordinates are
$$
x_{1,2}=-1\pm2\sqrt{-b}, \quad
y_{1,2}=\pm\sqrt{-b},\quad
x_{3,4}=1\pm2\sqrt{-b}, \quad
y_{3,4}=\mp\sqrt{-b}.
$$

We observe that the singular points $M_{1,2}$ are located on the hyperbola
and on the lines, whereas the singularities $M_{3,4}$ are located on
the invariant lines.
Moreover, at least one of the singular points $M_{3,4}$ could belong
to the hyperbola if an only if
$$
\big[\Phi(x_3,y_3)\big]\big[\Phi(x_4,y_4)\big]=4(b+1)(4b+1)=0,
$$
i.e. if and only if $4b+1=0$.

Since $x_1x_2=4(4b+1)$, the position of the singular points $M_{1,2}$
on the hyperbola is governed by $\operatorname{sign}(4b+1)$, while the position of
the invariant hyperbola is governed by $\operatorname{sign}(b)$.

We calculate
$$
\chi_A^{(8)}=-80\,b,\quad
\chi_D^{(8)}=80(4b+1),\quad
\mathcal{R}_9=40(b+1)
$$
and we conclude that $\operatorname{sign}(\chi_A^{(8)})=-\operatorname{sign}(b)$ and
$\operatorname{sign}(\chi_D^{(8)})=\operatorname{sign}(4b+1)$.

We observe that, if $b>0$, then all four singularities and the
invariant lines are complex. So, we arrive at the unique
configuration Config. H.79 if $\chi_A^{(8)}<0$.

In the case $\chi_A^{(8)}>0$, we get the following conditions and
configurations:
\begin{itemize}
\item $\mathcal{R}_9<0$ Config. H.96;

 \item $\mathcal{R}_9>0$ and $\chi_D^{(8)}<0$ $\Rightarrow$  Config. H.93;

 \item $\mathcal{R}_9>0$ and $\chi_D^{(8)}>0$ $\Rightarrow$  Config. H.92;

 \item $\mathcal{R}_9>0$ and $\chi_D^{(8)}=0$ $\Rightarrow$  Config. H.87.
\end{itemize}

If $\chi_A^{(8)}=0$, then $b=0$ and the invariant lines coalesce
and become a double line. Moreover, the singularity $M_1$ coalesces with
$M_3$, and so does $M_2$ with $M_4$, and we have two double
singular points, leading us to the configuration
Config. H.101.

\paragraph{Case $\beta_2=0$.} Then $d=0$ and, according to
Theorem \ref{trm:exist-hyp} (see Diagram \ref{diagr:eta-poz}) we
have at least one hyperbola if and only if the conditions
$(\mathfrak{C}_3)$ are satisfied, where by $(\mathfrak{C}_3)$ we denote
$$
(\mathfrak{C}_3):\
(\beta_1=0)\cap\big((\gamma_{12}=0,\mathcal{R}_9\ne0)\cup
(\gamma_{13}=0)\big)\big).
$$
We observe that the condition $\gamma_{12}=0$ leads to the
existence of only one invariant hyperbola, whereas the condition
$\gamma_{13}=0$ leads to the existence of an infinite number of
such hyperbolas.

We calculate\ $ \gamma_{12}=-128(a-4b)(4a-b),\quad
\gamma_{13}=4(a-b). $

\subparagraph{Subcase $\gamma_{12}=0$.} Then $(a-4b)(4a-b)=0$ and,
via the change $(x,y,a,b)\mapsto(y,x,b,a)$, we may assume $b=4a$
and we arrive at the 1-parameter family of systems
\begin{equation} \label{sys:CSb-2}
\frac {dx}{dt}= a+x^2,\quad
 \frac {dy}{dt}= 4a+y^2, \quad a\ne0.
\end{equation}
These systems possess two couples of parallel invariant lines and
the invariant hyperbola
\begin{equation} \label{Hyp:2b}
\begin{gathered}
\mathcal{L}_{1,2}(x,y)=x^2+a=0,\quad
\mathcal{L}_{3,4}(x,y)=y^2+4a=0,\\
\Phi(x,y)=a-x(x-y)=0.
\end{gathered}
\end{equation}

Since $B_1=B_2=0$ and $B_3=36ax^2y^2\ne0$, systems
\eqref{sys:CSb-2} could not possess other invariant lines rather
than the ones in \eqref{Hyp:2b}. Moreover as $\mu_0=1\ne0$, by
Lemma \ref{lem:mu_i-ISPs} the above systems possess finite
singularities $M_i(x_i,y_i)$ of total multiplicity four and
their coordinates are
$$
x_{1,2}=\pm\sqrt{-a}, \quad
y_{1,2}=\pm2\sqrt{-a},\quad
x_{3,4}=\pm\sqrt{-a}, \quad
y_{3,4}=\mp2\sqrt{-a}.
$$

We observe that all four singularities belong to the lines
$\mathcal{L}_{1,2,3,4}=0$. Moreover, the singular points
$M_{1,2}$ are located on the hyperbola, whereas the singular
points $M_{3,4}$ could not belong to the hyperbola becuase
$a\ne0$.

Since $x_1x_2=4a$, the position of the singular points $M_{1,2}$
on the hyperbola is governed by $\operatorname{sign}(a)$, as well as the
position of the invariant hyperbola.

We calculate $\chi_A^{(2)}=-80a $ and we conclude that
$\operatorname{sign}(\chi_A^{(2)})=-\operatorname{sign}(a)$.

Since in the case $a>0$ all four singularities and the invariant
lines are complex, we arrive at the configuration
Config. H.120 if $\chi_A^{(2)}<0$ and
Config. H.118 if $\chi_A^{(2)}>0$.

\subparagraph{Subcase $\gamma_{13}=0$.} Then $b=a$ and we arrive at the
1-parameter family of systems
\begin{equation} \label{sys:CSb-3}
\frac {dx}{dt}= a+x^2,\quad
 \frac {dy}{dt}= a+y^2,\quad a\ne0.
\end{equation}
These systems possess five invariant lines and the family of
invariant hyperbolas
\begin{equation} \label{Hyp:3b}
\begin{gathered}
\mathcal{L}_{1,2}(x,y)=x^2+a=0,\quad
\mathcal{L}_{3,4}(x,y)=y^2+a=0,\\
\mathcal{L}_5(x,y)=x-y=0,\quad
\Phi(x,y)=2a-r(x-y)+2xy=0, \quad r\in\mathbb{C}.
\end{gathered}
\end{equation}

Since $\mu_0=1\ne0$ the above systems possess finite
singularities $M_i(x_i,y_i)$ of total multiplicity four and
their coordinates are
$$
x_{1,2}=\pm\sqrt{-a}, \quad
y_{1,2}=\pm\sqrt{-a},\quad
x_{3,4}=\pm\sqrt{-a}, \quad
y_{3,4}=\mp\sqrt{-a}.
$$

We observe that all four singularities belong to the lines
$\mathcal{L}_{1,2,3,4}=0$. Moreover, the singular points $M_{1,2}$
are located on the hyperbolas for any $r\in\mathbb{C}$ and on the line
$\mathcal{L}_5=0$.

The $\operatorname{sign}(a)$ distinguishes if the singularities are either
real, or complex, or coinciding (if $a=0$). Since
$\mathcal{R}_9=16a$, we conclude that
$\operatorname{sign}(\mathcal{R}_9)=\operatorname{sign}(a)$.

In the case $a\ne0$, we could assume $a=1$ if $a>0$ and $a=-1$ if
$a<0$, by a rescaling. So, we arrive at the configuration
Config. H.160 if $\mathcal{R}_9<0$,
Config. H.161 if $\mathcal{R}_9>0$ and
Config. H.162 if $\mathcal{R}_9=0$.

The proof of statement (B) of Theorem \ref{th:Main1} is
complete.

\section{Configurations of invariant hyperbolas for the class QSH$_{(\eta=0)}$}

\begin{theorem}\label{th:Main2}
Consider the class {\rm QSH$_{(\eta=0)}$} of all
non-degenerate quadratic differential systems
\eqref{sys:QSgenCoef} possessing an invariant hyperbola and either exactly 
two distinct real
singularities at infinity or the line at infinity filled up with
singularities.

{\rm(A)}
This family is classified according to the configurations of invariant hyperbolas
 and of invariant straight lines of the systems, yielding 43 distinct such
 configurations.
 This geometric classification appears in  Diagrams \ref{diag:TMH=1}
 and  \ref{diag:TMH-ge2}. More precisely:
 \begin{itemize}
 \item[$(A1)$]
 There are exactly 9 configurations with an infinite number of invariant hyperbolas.
 \item[$(A2)$] The remaining 34 configurations could have up to a maximum
of 2 distinct invariant
 hyperbolas, real or complex, and up to 3 distinct invariant straight lines, real or
 complex, including the line at infinity.
 \end{itemize}

{\rm (B)}  Diagram \ref{diag:eta=0-chi} is the bifurcation diagram in the
space $\mathbb{R}^{12}$ of the coefficients of the system in {\rm QSH$_{(\eta=0)}$}
 according to their configurations of invariant hyperbolas and invariant
straight lines. Moreover,  Diagram \ref{diag:eta=0-chi} gives
an algorithm to compute the configuration of a system with an invariant
hyperbola for any quadratic differential system, presented in any normal form.
\end{theorem}

\begin{remark} \label{rmk12} \rm
In the above theorem we indicate that the 43 configurations obtained for the family
 {\rm QSH$_{(\eta=0)}$} are distinct because of
 the types of $ICD, ILD, MS_{0C}$ and PD. We defined in Section
\ref{sec:auxil results} such functions on the family {\rm QSH$_{(\eta=0)}$}.
We can read several geometrical invariants, modulo the group action,
from the expressions of these cycles. They form a complete set of geometric
invariants for the configurations of the family {\rm QSH$_{(\eta=0)}$}.
\end{remark}

\begin{remark} \label{rmk13} \rm
The invariant polynomials which appear in  Diagram~\ref{diagr:eta=0}
are introduced in Section \ref{sec:auxil results}. Moreover, in
this diagram we denote by $(\mathfrak{C}_1)$ the following condition
$$
 (\mathfrak{C}_1):  (\beta_6=0, \beta_{11}\mathcal{R}_{11}\ne0) \cap
 \big((\beta_{12}\ne0, \gamma_{15}=0)\cup (\beta_{12}=
 \gamma_{16}=0)\big).
$$
\end{remark}

\begin{remark}\label{rm:details-Main2}
For more details about the geometric classification of the configurations
of systems in {\rm QSH$_{(\eta=0)}$} see Section \ref{sec:conclusion}.
\end{remark}

\begin{figure}[ht]
\centering
\includegraphics[width=0.9\textwidth]{fig24}
 % Diagr_Geom_eta0_TMH-1-A.eps
\caption{Diagram of configurations with one simple
hyperbola} \label{diag:TMH=1}
\end{figure}

\begin{figure}[ht]
\centering
\includegraphics[width=0.9\textwidth]{fig25}
% Diagr_Geom_eta0_TMH-1-B.eps
\caption{(cont. of Diag. \ref{diag:TMH=1})
Configurations with one simple hyperbola}
\end{figure}

\begin{figure}[ht]
\centering
\includegraphics[width=0.9\textwidth]{fig26}
 % Diagr_Geom_eta0_TMH-ge2-v2.eps
\caption{Diagram of configurations with $TMH\ge2$} \label{diag:TMH-ge2}
\end{figure}

\begin{figure}[ht]
\centering
\includegraphics[width=0.9\textwidth]{fig27}
 % Diagram-Conditions-eta-zero-A1-v2.eps
\caption{Bifurcation diagram in $\mathbb{R}^{12}$ of the configurations:
 Case $\eta=0$} \label{diag:eta=0-chi}
\end{figure}

\begin{figure}[ht]
\centering
 \includegraphics[width=0.9\textwidth]{fig28}
 % Diagram-Conditions-eta-zero-B1-v2.eps
\caption{(cont. of Diag.\ref{diag:eta=0-chi}) Bifurcation
diagram in $\mathbb{R}^{12}$ of the configurations:
Case $\eta=0$}
\end{figure}


We prove part (A) under the assumption that part
(B) is already proved. Later, we prove part (B).
Summing up all the concepts introduced in order to define the
invariants, we end up with the list: (CD, ILD, MS$_{0C}$,
TMH, TML, PD, $O$ and $|\operatorname{Sing}_{\infty}|$.
The proof of part (A) of this theorem could be
done in a similar way of the proof of part (A) of Theorem \ref{th:Main1}.
\smallskip

\noindent\emph{Proof of part (B).} Following the conditions given by
Diagram \ref{diagr:eta=0} (the case $\eta=0$). We consider two
possibilities: $M(\tilde a,x,y)\ne0$ (i.e. at infinity we have two
distinct real singularities) and $M=0=C_2$ (when we have an
infinite number of singularities at infinity).

\subsection{Possibility $M(\tilde a,x,y)\ne0$}
According to Lemma \ref{lem:S1-S5} there exists a linear
transformation and time rescaling which brings systems
\eqref{sys:QSgenCoef} to the systems
\begin{equation} \label{sys:eta-zero-Gen}
\begin{gathered}
\frac {dx}{dt}= a+cx+dy+gx^2+hxy,\\
\frac {dy}{dt}= b+ex+fy+(g-1)xy+hy^2.
\end{gathered}
\end{equation}
For this systems we calculate
\begin{equation} \label{val:C2,theta-eta0}
C_2(x,y)=x^2y,\quad \theta= -h^2(g-1)/2.
\end{equation}

\subsubsection{Case $\theta\ne0$} In this case $h(g-1)\ne0$
and by a translation we may assume $d=e=0$. So in what follows
we consider the family of systems
\begin{equation} \label{sys:eta=0-theta-ne0}
\begin{gathered}
\frac {dx}{dt}= a+cx+ gx^2+hxy,\\
\frac {dy}{dt}= b+ fy+(g-1)xy+hy^2
\end{gathered}
\end{equation}
for which calculations yield:
$$
\gamma_1= (2c - f) (c + f)^2 h^4 (g-1)^2/32,\quad
\beta_2= h^2(2c - f)/2.
$$
According to Theorem \ref{trm:exist-hyp} for the existence of an
invariant hyperbola of the above systems the condition
$\gamma_1=0$ is necessary. So we consider two subcases:
$\beta_2\ne0$ and $\beta_2=0$.

\paragraph{Subcase $\beta_2\ne0$.} Then $2c-f\ne0$ and the condition
$\gamma_1=0$ implies $f=-c$. Then we calculate
\begin{gather*}
\gamma_2= -14175 c^2 h^5 (g-1)^2 (3g-1) [a(2g-1) - 2 b h],\\
\beta_1= -3c^2 h^2 (g-1) (3g-1)/4
\end{gather*}
and following  Diagram \ref{diagr:eta=0} (see Theorem
\ref{trm:exist-hyp}) we examine two possibilities: $\beta_1\ne0$
and $\beta_1=0$.

\subparagraph{Possibility $\beta_1\ne0$.} Then the necessary
condition $\gamma_2=0$ (for the existence of a hyperbola) gives
$a(2g-1) - 2 b h=0$ and setting $a=2a_1h$ (since $h\ne0$) we get
$b= a_1 (2g-1)$. Therefore keeping the old parameter $a$ (instead
of $a_1$) we arrive at the following family of systems
$$
 \frac {dx}{dt}= 2ah+cx +gx^2+hxy,\quad \frac {dy}{dt}= a(2g-1)
 -cy+(g-1)xy+hy^2.
$$
We observe that since $ch\ne0$ , we may assume $c=h=1$ by the
rescaling $(x,y,t)\mapsto (cx,cy/h,t/c)$ and the additional
parametrization $ah/c^2\to a$. So we get the following
2-parameter family of systems
\begin{equation} \label{sys:eta=0-theta-ne0-hyp-1}
 \frac {dx}{dt}= 2a + x +gx^2+ xy,\quad
 \frac {dy}{dt}= a(2g-1) - y+(g-1)xy+ y^2,
\end{equation}
which possess the following invariant hyperbola
(with cofactor $(2g-1)x+2y$):
\begin{equation} \label{Hyp:1bb}
 \Phi(x,y)= a + xy=0
\end{equation}
and for which the following coefficient conditions (defined by
$\theta\beta_2\beta_1\mathcal{R}_1\ne0$) must be satisfied:
\begin{equation} \label{cond:CS-1b}
a (g-1) (3g-1) \ne0.
\end{equation}
For systems \eqref{sys:eta=0-theta-ne0-hyp-1} we calculate
\begin{equation} \label{val:B1-CS1b}
B_1= 4 a^3 (g-1)^2 ( 1 - 2 g).
\end{equation}


(1) {\it Case $B_1\ne0$.} In this case by Lemma
\ref{lem:Bi-0} we have no invariant lines. For systems
\eqref{sys:eta=0-theta-ne0-hyp-1} we calculate\ $ \mu_0=g $\ and we
consider two subcases: $\mu_0\ne0$ and $\mu_0=0$.


(a) {\it Subcase $\mu_0\ne0$.} Then by Lemma
\ref{lem:mu_i-ISPs} the systems have finite singularities of
total multiplicity four. More exactly, systems
\eqref{sys:eta=0-theta-ne0-hyp-1} possess the singular points
$M_{1,2}\big(x_{1,2} ,y_{1,2} \big)$ and $M_{3,4}\big(x_{3,4}
,y_{3,4} \big)$, where
\begin{gather*}
x_{1,2}=\frac{-1 \pm \sqrt{1-4ag}}{2g},\quad
y_{1,2}= \frac{1 \pm \sqrt{1-4ag}}{2},\\
x_{3,4}= -1 \pm \sqrt{1-4a},\quad y_{3,4}
= (2g-1) (1 \mp \sqrt{1-4a}\,)/2.
\end{gather*}
We detect that the singularities
 $M_{1,2}\big(x_{1,2}, y_{1,2} \big)$ are located on the hyperbola.
On the other hand for systems
\eqref{sys:eta=0-theta-ne0-hyp-1} we calculate the invariant
polynomials
$$
\chi_A^{(1)}=9(g-1)^2 (3g-1)^2(1-4ag)/64
$$
and by \eqref{cond:CS-1} we conclude that
$\operatorname{sign}(\chi_A^{(1)})=\operatorname{sign}(1-4ag)$ (if $1-4ag\ne0$) and we
consider three possibilities: $\chi_A^{(1)}<0$, $\chi_A^{(1)}>0$
and $\chi_A^{(1)}=0$.


(a1) {\it Possibility $\chi_A^{(1)}<0$.} So we
have no real singularities located on the invariant hyperbola and
we arrive at the configurations of invariant curves given by
Config. $\widetilde{{\rm H}}.1$.

(a2) {\it Possibility $\chi_A^{(1)}>0$.} In this
case the singularities $M_{1,2}\big(x_{1,2}, y_{1,2} \big)$
located on the hyperbola are real and we have the next result.

\begin{lemma} \label{lem:sign(x1*x2)}
Assume that the singularities $M_{1,2}\big(x_{1,2} ,y_{1,2} \big)$
(located on the hyperbola) are finite. Then these singularities
are located on different branches of the hyperbola if
$\chi_C^{(1)}<0$ and they are located on the same branch if
$\chi_C^{(1)}>0$, where $\chi_C^{(1)}=315ag (g-1)^4(3g-1)^2/32$.
\end{lemma}

\begin{proof} Since the asymptotes of the hyperbola \eqref{Hyp:1} are
the lines $x=0$ and $y=0$ it is clear that the singularities
$M_{1,2}$ are located on different branches of the hyperbola if and
only if $x_1x_2<0$. We calculate
\begin{equation} \label{val:chi-C(1)}
\begin{gathered}
x_1x_2=\big[\frac{-1+ \sqrt{1-4ag}}{2g}\big] \big[\frac{-1 - \sqrt{1-4ag}}{2g}\big]
=\frac{a}{g}
\end{gathered}
\end{equation}
and because of condition \eqref{cond:CS-1} we obtain that
$\operatorname{sign}(x_1x_2)=\operatorname{sign}(\chi_C^{(1)})$. This completes the proof of the
lemma.
\end{proof}


Other two singular points $M_{3,4}\big(x_{3,4} ,y_{3,4} \big)$ of
systems \eqref{sys:eta=0-theta-ne0-hyp-1} are generically located
outside the hyperbola. We need to determine the conditions when
some singular points of the system become singular points lying
on the hyperbola. Considering \eqref{Hyp:1} we calculate
$$
\Phi(x,y)|_{\{x=x_{3,4},\ y=y_{3,4}\}}= (2g-1) \big(-1\pm
\sqrt{1-4a}\big)+a(4g-1) \equiv \Omega_{\pm}(a,g).
$$
Put $\Omega_3(a,g)=\Omega_{+}(a,g)$ and $\Omega_4(a,g)=\Omega_{-}(a,g)$.
It is clear that at least one of the singular points
$M_3(x_3,y_3)$ or $M_4(x_4,y_4)$ belongs to the hyperbola
\eqref{Hyp:1} if and only if
$$
\Omega_3 \Omega_4= a\big[2(1-2g)+a(1-4g)^2\big]\equiv a Z_1=0.
$$
On the other hand for systems \eqref{sys:eta=0-theta-ne0-hyp-1} we
have\ $ \widetilde{\,\chi}_D^{(1)}=54 Z_1 $\ and clearly from
\eqref{cond:CS-1} the condition $\widetilde{\,\chi}_D^{(1)}=0$ is
equivalent to $Z_1=0$. We examine two cases:
$\widetilde{\,\chi}_D^{(1)}\ne0$ and
$\widetilde{\,\chi}_D^{(1)}=0$.


($\alpha$) {\it Case $\widetilde{\,\chi}_D^{(1)}\ne0$.}
Then $Z_1\ne0$ and on the
hyperbola there are two simple real singularities (namely $M_{1,2}
(x_{1,2} ,y_{1,2})$). By Lemma \ref{lem:sign(x1*x2)} their
position is defined by the invariant polynomial $\chi_C^{(1)}$ and
we arrive at the configuration given by
Config. $\widetilde{{\rm H}}.6$ if $\chi_C^{(1)}<0$ and by
Config. $\widetilde{{\rm H}}.5$ if $\chi_C^{(1)}>0$.

($\beta$) {\it Case $\widetilde{\,\chi}_D^{(1)}=0$.}
In this case the condition
$Z_1=0$ implies $4g-1\ne0$ (otherwise for $g=1/4$ we get
$Z_1=1\ne0$. So we obtain $a=2(2g-1)/(4g-1)^2$. In this case the
coordinates of the finite singularities $M_i(x_i,y_i)$ ($i=1,2,3,4$)
are as follows
\begin{gather*}
x_1=\frac{1-2g}{g(4g-1)},\quad y_1= \frac{2g}{4g-1}; \quad
x_2=x_3=\frac{2}{1-4g},\\
 y_2=y_3= \frac{2g-1}{4g-1},\quad
x_4=\frac{4(1-2g)}{(4g-1},\quad y_4= \frac{2(g-1)^2}{4g-1},
\end{gather*}
i.e. all the singularities are real. Then considering
Proposition~\ref{prop:number-FSPs} we calculate
\begin{gather*}
{\rm D}=0,\quad {\rm T}=-3\big[2g (g-1 ) x + (2g-1) y\big]^2{\rm P},\\
{\rm P}= \frac{(4g-3)^2 (g x - y)^2 (2 g x-x + 2 y)^2}{(4g-1)^4}.
\end{gather*}

($\beta1$) {\it Subcase ${\rm T}\ne0$.} Then ${\rm T}<0$
and according to Proposition \ref{prop:number-FSPs} systems
\eqref{sys:eta=0-theta-ne0-hyp-1} possess one double and two simple real finite
singularities. As it is mentioned above, the singular point
$M_3(x_3,y_3)$ coalesces with the singular point $M_2(x_2,y_2)$
located on the hyperbola, whereas $M_4(x_4,y_4)$ remains outside
the hyperbola.

Considering the coordinates of the singular points we calculate
$$
\operatorname{sign}(x_1x_2)= \operatorname{sign}(g(2g-1)),\quad
\chi_C^{(1)}=\frac{315g(2g-1)(g-1)^4(3g-1)^2}{16(4g-1)^2}.
$$
Therefore in the case $\chi_C^{(1)}<0$ the singular points $M_1$
and $M_2=M_3$ are located on different branches of the hyperbola
and we arrive at the configuration Config. $\widetilde{{\rm H}}.10$.

Assume now that the condition $\chi_C^{(1)}>0$ holds, i.e. the two
singular points (one double and one simple) are located on the
same branch of the hyperbola. Since on this branch are also
located two infinite singular points (one double and one simple),
it is clear that the reciprocal position of singular points
$M_1$ and $M_2$ (double) on the branch leads do different
configurations. So we need to determine the conditions to
distinguish these two situations.

We calculate
$$
x_1-x_2=\frac{1-2g}{g(4g-1)}-\frac{2}{1-4g}=\frac{1}{g(4g-1)}
$$
and hence the reciprocal position of $M_1$ and $M_2$ depends on
the sign of the expression $g(4g-1)$. On the other hand, the
condition $\chi_C^{(1)}>0$ implies $g(2g-1)>0$, i.e. we have
either $g<0$ or $g>1/2$. Since $\mu_0=g$ we deduce that these two
possibilities are governs by the invariant polynomial $\mu_0$.

It is easy to detect that we arrive at
Config. $\widetilde{{\rm H}}.8$ if $\mu_0<0$ (i.e. $g<0$) and
we get Config. $\widetilde{{\rm H}}.9$ if $\mu_0>0$ (i.e.
$g>1/2$).

($\beta2$) {\it Subcase ${\rm T}=0$.} In this case
from the condition $B_1\ne0$ (i.e. $2g-1\ne0$) the equality
${\rm T}=0$ holds if and only if ${\rm P}=0$ which is equivalent to
$4g-3=0$, i.e. $g=3/4$. In this case we obtain
$$
 {\rm D}={\rm T}={\rm P}=0,\quad {\rm R}=3(3 x - 4 y)^2/64
$$
and since ${\rm R}\ne0$, by Proposition \ref{prop:number-FSPs} we
obtain one triple and one simple singularities. More precisely the
singular points $M_2$, $M_3$ and $M_4$ coalesce and since all the
parameters of systems \eqref{sys:eta=0-theta-ne0-hyp-1} are fixed
we get the unique configuration given by
Config. $\widetilde{{\rm H}}.11$.


(a3) {\it Possibility $\chi_A^{(1)}=0$.} In this
case we get $g=1/(4a)$ and the singularities $M_{1,2}\big(x_{1,2},
y_{1,2} \big)$ located on the hyperbola coincide. On the other
hand we have $Z_1=a\ne0$ and hence none of the singular points
$M_{3,4}$ could belong to the hyperbola. So we arrive at the
unique configuration presented by Config. $\widetilde{{\rm H}}.4$.


(b) {\it Subcase $\mu_0=0$.} Then we have $\mu_1=-y$
and by Lemma \ref{lem:mu_i-ISPs} one finite singular point has
gone to infinity and coalesced with the infinite singular point
$[1,0,0]$. In this case we arrive at the 1-parameter family of
systems
\begin{equation} \label{sys:eta=0-theta-ne0-hyp-1a}
 \frac {dx}{dt}= 2a + x + xy,\quad
 \frac {dy}{dt}= - a - y-xy+ y^2
\end{equation}
possessing the singular points $M_1'(x_1',y_1')$ and
$M_{2,3}(x_{2,3},y_{2,3})$ (the same points for the particular
case $g=0$) with the coordinates
$$
x_1'=-a, \quad y_1'=1;\quad x_{3,4}= -1 \pm \sqrt{1-4a},\quad
y_{3,4}= (-1 \pm \sqrt{1-4a}\,)/2.
$$
We observe that only the singular point $M_1'$ is located on the
hyperbola. On the other hand it was shown earlier that one of
the points $M_{2,3}(x_{2,3},y_{2,3})$ belongs to the hyperbola if
and only if $Z_1=0$ which in this case gets the value $Z_1=a+2$.
For systems \eqref{sys:eta=0-theta-ne0-hyp-1a} we calculate
$$
\widetilde{\chi}_D^{(1)}=54(a+2)
$$
and it is not too difficult to detect that in the case
$\widetilde{\,\chi}_D^{(1)}\ne0$ (i.e. $a+2\ne0$) we arrive at the
unique configuration given by Config. $\widetilde{{\rm H}}.3$.

Assume now $\widetilde{\chi}_D^{(1)}=0$. Then $a=-2$ and we get
a system with constant coefficients for which the singular point
$M_2$ has coalesced with $M_1'$. As a result we obtain
Config. $\widetilde{{\rm H}}.7$.



(2) {\it Case $B_1=0$.} Considering \eqref{val:B1-CS1}
and the condition \eqref{cond:CS-1} this implies $g=1/2$ and we
obtain the following 1-parameter family of systems
\begin{equation} \label{sys:eta=0-theta-ne0-hyp-1b}
 \frac {dx}{dt}=2 a + x + x^2/2 + x y,\quad
 \frac {dy}{dt}= -y (1 + x/2 - y).
\end{equation}
These systems besides the hyperbola \eqref{Hyp:1} possess the
invariant line $y=0$ and four singular points $M_i(x_i,y_i)$ with
the coordinates
\begin{gather*}
x_{1,2}= -1 \pm \sqrt{1-2a},\quad
 y_{1,2}= \frac{1 \pm \sqrt{1-2a}}{2},\\
x_{3,4}= -1 \pm \sqrt{1-4a},\quad y_{3,4}=0.
\end{gather*}
We observe that the singular point $M_1$ and $M_2$ are located on
the hyperbola, whereas $M_3$ and $M_4$ are situated on the
invariant line $y=0$, which is one of the asymptotes of the
hyperbola \eqref{Hyp:1}. For the above systems we calculate
$$
{\rm D}= 48 a^2 ( 1 - 2 a) (4a-1), \quad
 \chi_A^{(1)}=9(1-2a)/1024
$$
and it is clear that from the condition \eqref{cond:CS-1} (i.e.
$a\ne0$) two of the finite singular point could coalesce if and
only if ${\rm D}=0$. So we examine three subcases: ${\rm D}<0$, ${\rm D}>0$
and ${\rm D}=0$.


(a) {\it Subcase ${\rm D}<0$.} Then $( 1 - 2 a)
(4a-1)<0$ and we observe that if $\chi_A^{(1)}<0$ (i.e. $a>1/2$)
all the singular points are complex and we get the unique
configuration given by Config. $\widetilde{{\rm H}}.12$.

Assume now $\chi_A^{(1)}>0$ (i.e. $a<1/2$). Then the condition
${\rm D}<0$ implies $a<1/4$ and all singular points are real. We
calculate $ x_1x_2=2a$ and $\chi_C^{(1)}=316a/4096$ and hence
this invariant polynomials governs the position of the singular
points located on the hyperbola (on the same branch or not). Thus
we get Config. $\widetilde{{\rm H}}.17$ when $\chi_C^{(1)}<0$
and Config. $\widetilde{{\rm H}}.16$ when $\chi_C^{(1)}>0$.


(b) {\it Subcase ${\rm D}>0$.} In this case we have
$1/4<a<1/2$ and therefore the singular points located on the
hyperbola are real, whereas the singularities from the invariant
line are complex. As $a>0$ we deduce that the real singularities
are located on the same branch of the hyperbola. As a result, we
get the unique configuration Config. $\widetilde{{\rm H}}.14$.



(c) {\it Subcase ${\rm D}=0$.} Then either $a=1/4$ or
$a=1/2$ and these possibilities are distinguished by
$\chi_A^{(1)}$. Therefore we get the configuration
Config. $\widetilde{{\rm H}}.15$ if $\chi_A^{(1)}\ne0$ and
Config. $\widetilde{{\rm H}}.13$ if $\chi_A^{(1)}=0$.


\subparagraph{Possibility $\beta_1=0$.} Then because
$\theta\ne0$ (i.e. $h(g-1)\ne0$) and to the condition $\beta_2= 3
c h^2/2\ne0$, the condition $\beta_1=0$ implies $g=1/3$ and
$\gamma_2=0$. So we arrive at the following family of systems
$$
 \frac {dx}{dt}= a + c x + x^2/3 + h x y,\quad
 \frac {dy}{dt}= b - c y - 2 x y/3 + h y^2.
$$
We observe that since $ch\ne0$ we may assume $c=h=1$ by the
rescaling $(x,y,t)\mapsto (cx,cy/h,t/c)$. According to Theorem
\ref{trm:exist-hyp} (see Diagram \ref{diagr:eta=0}) the above systems
possess an invariant hyperbola if and only if $\gamma_4=0$ and
$\mathcal{R}_3\ne0$. Considering the condition $c=h=1$ for these
systems we calculate
$$
\gamma_4=16 (a + 6 b)^2/3,\quad \mathcal{R}_3=3 b/2
$$
and hence the condition $\gamma_4=0$ gives $b=-a/6\ne0$. So we get
the following 1-parameter family of systems
\begin{equation} \label{sys:eta=0-theta-ne0-hyp-2}
 \frac {dx}{dt}= a + x + x^2/3 + x y,\quad
 \frac {dy}{dt}= -a/6 - y - 2 x y/3 + y^2
\end{equation}
with $a\ne0$ which possess the following invariant hyperbola
\begin{equation} \label{Hyp:2}
 \Phi(x,y)= a + 2xy=0
\end{equation}
and singular points $M_i(x_i,y_i)$ (i=1,2,3,4) with the
coordinates
\begin{gather*}
x_{1,2}=(-3 \pm \sqrt{3(3-2a)}/2,\quad
y_{1,2}= (3 \pm \sqrt{3(3-2a)}/6,\\
x_{3,4}= -1 \pm \sqrt{1-2a},\quad y_{3,4}= (-1 \pm
\sqrt{1-2a}\,)/6.
\end{gather*}
We observe that the singularities $M_{1,2}\big(x_{1,2}, y_{1,2}\big)$
are located on the hyperbola and since
$\chi_A^{(2)}=2(3-2a)/9$ we deduce that these points are complex
(respectively, real) if $\chi_A^{(2)}<0$ (respectively
$\chi_A^{(2)}>0$) and they coincide if $\chi_A^{(2)}=0$.

On the other hand we have $x_1x_2=3 a/2$ and $\chi_A^{(8)}=23
a/12$ and therefore we conclude that the singular points $M_{1,2}$
are located on different branches of the hyperbola if
$\chi_A^{(8)}<0$ and on the same branch if $\chi_A^{(8)}>0$.

Other two singular points $M_{3,4}\big(x_{3,4} ,y_{3,4} \big)$ of
systems \eqref{sys:eta=0-theta-ne0-hyp-2} generically are located
outside the hyperbola. In order to determine the conditions when
at least one of these singular points is located on the hyperbola
we calculate
\begin{gather*}
\Phi(x,y)|_{\{x=x_{3,4},\ y=y_{3,4}\}}= (a+2 \mp 2 \sqrt{1 - 2
a})/3\equiv \Omega_{3,4}(a),\\
 \Omega_3 \Omega_4= a (12 + a)/9,\quad \chi_E^{(3)}=-9 a (12 + a)/8.
 \end{gather*}
It is clear that at least one of the singular points $M_3$ or
$M_4$ belongs to the hyperbola
\eqref{Hyp:2} if and only if $\chi_E^{(3)}=0$.

Since for systems \eqref{sys:eta=0-theta-ne0-hyp-2} we have $B_1=2
a^3/27\ne0$ and $\mu_0=1/3\ne0$, by Lemmas \ref{lem:mu_i-ISPs}
and \ref{lem:Bi-0} we have no invariant lines and none of the
finite singularities could go to infinity. So we arrive at the
following conditions and configurations:
\begin{itemize}
 \item $\chi_A^{(2)}<0$ $\Rightarrow$  Config. $\widetilde{{\rm H}}.1$;
 \item $\chi_A^{(2)}>0$, $\chi_A^{(8)}<0$ and $\chi_E^{(3)}\ne0$ $\Rightarrow$
  Config. $\widetilde{{\rm H}}.6$;
 \item $\chi_A^{(2)}>0$, $\chi_A^{(8)}<0$ and $\chi_E^{(3)}=0$ $\Rightarrow$
 Config. $\widetilde{{\rm H}}.10$;
 \item $\chi_A^{(2)}>0$ and $\chi_A^{(8)}>0$  $\Rightarrow$
 Config. $\widetilde{{\rm H}}.5$;
 \item $\chi_A^{(2)}=0$ $\Rightarrow$  Config. $\widetilde{{\rm H}}.4$.
\end{itemize}

\paragraph{Subcase $\beta_2=0$.} Then $f=2c$ and this implies
$\gamma_1=0$. By Theorem \ref{trm:exist-hyp} (see  Diagram
\ref{diagr:eta=0}) in this case we have an invariant hyperbola if
and only if $\gamma_2=\beta_1=\gamma_{14}=0$ and
$\mathcal{R}_{10}\ne0$. Moreover, this hyperbola is simple if
$\beta_7\beta_8\ne0$ and it is double if $\beta_7\beta_8=0$. So we
calculate
$$
\gamma_2= -14175 a c^2 h^5 (g-1)^3 (1 + 3 g),\quad \beta_1=-9 c^2 (g-1)^2 h^2/16
$$
and evidently the condition $\gamma_2=\beta_1=0$ implies $c=0$.
Then we obtain
$$
\gamma_{14}=-80 h^3 \big[a(2g-1) - 2 b h\big],\quad
\mathcal{R}_{10}=-4 a h^2\ne0
$$
and as $h\ne0$ the condition $\gamma_{14}=0$ gives $a(2g-1) - 2 b
h=0$. Then setting $a=2a_1h$ we get $b= a_1 (2g-1)$ and keeping
the old parameter $a$ (instead of $a_1$) after the additional
rescaling $y\to y/h$ we arrive and at the following 2-parameter
family of systems
\begin{equation} \label{sys:eta=0-theta-ne0-hyp-3}
 \frac {dx}{dt}= 2a +gx^2+ xy,\quad \frac {dy}{dt}= a(2g-1)
 +(g-1)xy+ y^2.
\end{equation}
These systems possess the invariant hyperbola \eqref{Hyp:1} and we
calculate
$$
 \beta_7=8 (1 - 2 g),\quad
\beta_8= 32 (1 - 4 g),\quad B_1= 4 a^3 (g-1)^2 (1 - 2 g),\quad
\mu_0=g
$$
and following  Diagram \ref{diagr:eta=0} (see Theorem
\ref{trm:exist-hyp}) we examine two possibilities:
$\beta_7\beta_8\ne0$ and $\beta_7\beta_8=0$.

\subparagraph{Possibility $\beta_7\beta_8\ne0$.} In this case
for systems \eqref{sys:eta=0-theta-ne0-hyp-3} the condition
\begin{equation} \label{cond:CS-2a}
a(g-1)(2g-1)(4g-1)\ne0
\end{equation}
is satisfied and this implies $B_1\ne0$. Therefore according to
Lemma \ref{lem:Bi-0} these systems could not have invariant lines
and as earlier we consider two cases: $\mu_0\ne0$ and $\mu_0=0$.


(1) {\it Case $\mu_0\ne0$.} Then systems
\eqref{sys:eta=0-theta-ne0-hyp-3} possess four finite singular
points $M_i(x_i,y_i)$ (i=1,2,3,4) with the coordinates
\begin{gather*}
x_{1,2}= \pm \sqrt{-a/g},\quad y_{1,2}= \pm \sqrt{-ag},\\
x_{3,4}= \pm 2\sqrt{-a},\quad y_{3,4}= \pm \sqrt{-a}(1-2g).
\end{gather*}
We detect that the singularities $M_{1,2}\big(x_{1,2}, y_{1,2}
\big)$ are located on the hyperbola and they are complex
(respectively, real) if $ag>0$ (respectively $ag<0$). Moreover
since $x_1x_2=a/g$ then in the case when they are real (i.e.
$ag<0$) these points are located on different branches of the
hyperbola
\eqref{Hyp:1}.

On the other hand considering singular points
$M_{3,4}\big(x_{3,4}, y_{3,4} \big)$ we calculate
$$
\Phi(x,y)|_{\{x=x_3,\ y=y_3\}}=\Phi(x,y)|_{\{x=x_4,\
y=y_4\}}= a (4g-1)\ne0,
$$
i.e. for any values of the parameters $a$ and $g$ satisfying the
condition \eqref{cond:CS-2a} these singularities could not belong
to the hyperbola \eqref{Hyp:1}.

For systems \eqref{sys:eta=0-theta-ne0-hyp-3} we calculate
$\mu_0\mathcal{R}_{10}=-8ag\ne0$ and hence
$\operatorname{sign}(\mu_0\mathcal{R}_{10})=-\operatorname{sign}(ag)$. So we arrive at the
configuration given by Config. $\widetilde{{\rm H}}.1$ if
$\mu_0\mathcal{R}_{10}<0$ and by Config. $\widetilde{{\rm H}}.6$
if $\mu_0\mathcal{R}_{10}>0$.

(2) {\it Case $\mu_0=0$.} Then $g=0$ and we calculate
$$
\mu_0=\mu_1=0,\quad \mu_2=ay^2\ne0
$$
and by Lemma \ref{lem:mu_i-ISPs} two finite singular points
have gone to infinity and both coalesced with the infinite
singular point $[1,0,0]$. As a result we get the unique
configuration Config. $\widetilde{{\rm H}}.2$.


\subparagraph{Possibility $\beta_7\beta_8=0$.} Assume first
$\beta_7=0$, i.e. $g=1/2$ which implies $B_1=0$ and systems
\eqref{sys:eta=0-theta-ne0-hyp-3} possess the invariant line
$y=0$. Since $\mathcal{R}_{10}=-8a$, considering the coordinates
of the singularities we arrive at Config. $\widetilde{{\rm H}}.31$
if $\mathcal{R}_{10}<0$ and at Config. $\widetilde{{\rm H}}.32$
if $\mathcal{R}_{10}>0$.

Suppose now $\beta_8=0$ which gives $g=1/4$. Then the
singularities $M_3$ and $M_4$ coalesce with $M_1$ and $M_2$,
respectively. So in this case systems
\eqref{sys:eta=0-theta-ne0-hyp-3} have two double singular points located
on the hyperbola which are complex if $a>0$ and real if $a<0$. So
we obtain Config. $\widetilde{{\rm H}}.29$ if
$\mathcal{R}_{10}<0$ and Config. $\widetilde{{\rm H}}.30$ if
$\mathcal{R}_{10}>0$.


\subsubsection{Case $\theta=0$} According to
\eqref{val:C2,theta-eta0} we get $h(g-1)=0$ and since for systems
\eqref{sys:eta-zero-Gen} we have $\mu_0=gh^2$ we consider two
subcases: $\mu_0\ne0$ and $\mu_0=0$.

\paragraph{Subcase $\mu_0\ne0$.} Then $h\ne0$ and the condition
$\theta=0$ yields $g=1$. Since $h\ne0$ via the affine
transformation
$$
x_1=x+d/h,\quad y_1=hy+c-2d/h
$$
we may assume $d=f=0$, $h=1$ and systems
\eqref{sys:eta-zero-Gen} become as systems
\begin{equation} \label{sys:eta=theta=0-gen}
 \frac {dx}{dt}= a + cx+ x^2 + x y,\quad
 \frac {dy}{dt}= b +ex + y^2
\end{equation}
for which we calculate
$$
N=9y^2,\quad \beta_4=2,\quad \beta_3=-e/4,\quad \gamma_1= 9 ce^2/16.
$$

Since $N\beta_4\ne0$ following Diagram \ref{diagr:eta=0} (see
Theorem \ref{trm:exist-hyp}) for the existence of an invariant
hyperbola the conditions $\gamma_1=\gamma_2=\beta_3=0$ are
necessary. Therefore we have $e=0$ and this implies
$\gamma_1=\gamma_2=0$ and
$$
\gamma_8=42 (9 a - 18 b -2 c^2)^2.
$$
So setting for simplicity $c=3c_1$ and $a=2a_1$ the condition
$\gamma_8=0$ yields $b=a_1-c_1^2$ and keeping the notation for the
parameters $c$ and $a$ we arrive at the 2-parameter family of
systems
\begin{equation} \label{sys:eta=theta=0-hyp-1}
 \frac {dx}{dt}= 2 a + 3 c x + x^2 + x y,\quad
 \frac {dy}{dt}= a-c^2 + y^2.
\end{equation}
These systems possess the following invariant hyperbola and two
invariant lines:
\begin{equation} \label{Hyp:3}
 \Phi(x,y)= a+cx+ xy=0,\quad L_{1,2}=y\pm\sqrt{c^2-a}=0
\end{equation}
and singular points $M_i(x_i,y_i)$ (i=1,2,3,4) with the
coordinates
\begin{gather*}
x_{1,2}= -c \pm \sqrt{c^2-a},\quad y_{1,2}= \pm\sqrt{c^2-a} ,\\
x_{3,4}= -2(c \pm \sqrt{c^2-a}),\quad y_{3,4}= \pm\sqrt{c^2-a}.
\end{gather*}
The singularities $M_{1,2}\big(x_{1,2}, y_{1,2} \big)$ are
located at the intersection points of the hyperbola with invariant
lines, whereas the singularities $M_{3,4}$ are located only on
the invariant lines. More precisely, the singular point $M_3$
(respectively, $M_4$) is located on the same invariant line as the
singularity $M_1$ (respectively, $M_2$). Since
$\chi_A^{(7)}=(c^2-a)/4$ we deduce that all these finite singular
points as well as the invariant lines $L_{1,2}$ are complex
 if $\chi_A^{(7)}<0$ and real if
$\chi_A^{(7)}>0$. In the case $\chi_A^{(7)}=0$ (then $a=c^2\ne0$)
we obtain that the singular point $M_1$ (respectively, $M_3$)
coincides with $M_2$ (respectively, $M_4$) and moreover, in this
case invariant lines coincide, too. So we consider three
possibilities: $\chi_A^{(7)}<0$, $\chi_A^{(7)}>0$ and
$\chi_A^{(7)}=0$.

\subparagraph{Possibility $\chi_A^{(7)}<0$.} Then $c^2-a<0$
(this implies $a>0$) and all the singularities and the invariant
lines are complex. As a result we arrive at the unique
configuration given by Config. $\widetilde{{\rm H}}.18$.

\subparagraph{Possibility $\chi_A^{(7)}>0$.} In this case the
finite singularities $M_1\ne M_2$ and $M_3\ne M_4$ are real and we
observe that the singular points $M_{3,4}$ of systems
\eqref{sys:eta=theta=0-hyp-1} generically are located outside
the hyperbola. We calculate
\begin{gather*}
\Phi(x,y)|_{\{x=x_{3,4},\quad
 y=y_{3,4}\}}= 3a+4c(-c \pm 2
\sqrt{c^2-a})\equiv \Omega_{3,4}(a,c),\\
 \Omega_3 \Omega_4= a (9 a - 8 c^2).
\end{gather*}
On the other hand by Theorem \ref{trm:exist-hyp} (see  Diagram
\ref{diagr:eta=0}) the hyperbola \eqref{Hyp:3} is simple if
$\delta_4=3 (9 a - 8 c^2)\ne0$ and it is double if $\delta_4=0$.
So we conclude that at least one of the singularities $M_{3,4}$
belongs to the hyperbola if and only if the hyperbola is double
(i.e. when $\delta_4=0$). So we consider two cases: $\delta_4\ne0$
and $\delta_4=0$.


(1) {\it Case $\delta_4\ne0$.} Then all four finite
singularities are real and distinct. In this case in order to
detect the different configurations we need to distinguish the
position of the branches of the hyperbola (which depends on the
sign of the parameter $a$) as well as the position of the singular
point $M_3$ on the line $y=\sqrt{c^2-a}$ with respect to $M_1$ and
the position of $M_4$ on the line $y=-\sqrt{c^2-a}$ with respect
to $M_2$. So considering the coordinates of the finite
singularities we calculate
$$
x_1x_2=a,\quad (x_1-x_3)(x_2-x_4)=9 a - 8 c^2,\quad
\mathcal{R}_7=-3 a/4, \quad \chi_F^{(7)}=9 a - 8 c^2.
$$
So the singularities $M_1$ and $M_2$ are located on the same
branch of the hyperbola if $\mathcal{R}_7<0$ and on different
branches if $\mathcal{R}_7>0$. To determine exactly the position
of $M_1$ and $M_3$ as well as of $M_2$ and $M_4$ we observe, that
by the rescaling $(x,y,t)\mapsto (-x,-y,-t)$ we may assume
that the parameter $c\ge0$. This means that $x_1-x_3=c + 3
\sqrt{c^2-a}>0$ (because $c\ge0$ and $c^2-a>0$) and hence the sign
of $x_2-x_4$ is governed by the invariant polynomial
$\chi_F^{(7)}$.

Thus in the case $\chi_A^{(7)}>0$ and $\delta_4\ne0$ (then
$\chi_F^{(7)}\ne0$) we arrive at the following conditions and
configurations:
\begin{itemize}
 \item $\mathcal{R}_7<0$ and $\chi_F^{(7)}<0$  $\Rightarrow$  Config.
  $\widetilde{{\rm H}}.22$;
 \item $\mathcal{R}_7<0$ and $\chi_F^{(7)}>0$  $\Rightarrow$  Config.
  $\widetilde{{\rm H}}.20$;
 \item $\mathcal{R}_7>0$ $\Rightarrow$  Config. $\widetilde{{\rm H}}.21$.
\end{itemize}

(2) {\it Case $\delta_4=0$.} Then $a=8c^2/9\ne0$ and
by Theorem \ref{trm:exist-hyp} (see  Diagram \ref{diagr:eta=0}) the
hyperbola \eqref{Hyp:3} is double. Moreover in this case the
singular point $M_4$ coincides with $M_2$, located on the
hyperbola. Since $c\ne0$ (i.e. no other singularities could
coincide) we get the unique configuration
Config. $\widetilde{{\rm H}}.33$.

\subparagraph{Possibility $\chi_A^{(7)}=0$.} Then $a=c^2\ne0$
and this implies the coalescense of the singularity $M_2$ with
$M_1$ and of $M_4$ with $M_3$. Clearly in this case we get the
double invariant line $y=0$ and since $c\ne0$ we obtain
Config. $\widetilde{{\rm H}}.23$.


\paragraph{Subcase $\mu_0=0$.} Then the condition $\theta=\mu_0=0$
gives $h=0$ and for systems \eqref{sys:eta-zero-Gen} in this
case we calculate
$$
N=9(g-1) (1 + g) x^2,\quad \gamma_1=\gamma_2=\beta_4=0,\quad
\beta_6= d (g-1) (1 + g)/4.
$$
We next consider two possibilities: $N\ne0$ and $N=0$.

\subparagraph{Possibility $N\ne0$.} In this case by Theorem
\ref{trm:exist-hyp} (see  Diagram \ref{diagr:eta=0}) for the
existence of at least one hyperbola the condition $(\mathfrak{C}_1)$
are necessary and sufficient, where
$$
 (\mathfrak{C}_1): \ (\beta_6=0, \beta_{11}\mathcal{R}_{11}\ne0) \cap
 \big((\beta_{12}\ne0, \gamma_{15}=0)\cup (\beta_{12}= \gamma_{16}=0)\big).
$$
So the condition $\beta_6= 0$ is necessary. Since $N\ne0$ we get
$d=0$ and moreover as $g-1\ne0$, due a translation, we may assume
$e=f=0$. Therefore we arrive at the family of systems
$$
 \frac {dx}{dt}= a + c x + g x^2 ,\quad
 \frac {dy}{dt}= b +(g-1)x y,
$$
for which following Diagram \ref{diagr:eta=0} we calculate:
\begin{gather*}
\beta_{11}=4 (2g-1) x^2,\quad \mathcal{R}_{11}= -3 b (g-1)^2 x^4,\quad
\beta_{12}=(3g-1)x, \\
\gamma_{15}= 4(g-1)^2 (3g-1) \big[a (3g-1)^2 + c^2 (1-2g)\big]x^5.
\end{gather*}
So according to Theorem \ref{trm:exist-hyp} the condition
$\beta_{11}\mathcal{R}_{11}\ne0$ is necessary for the existence of
a hyperbola and considering  Diagram \ref{diagr:eta=0} we have to
consider the two cases: $\beta_{12}\ne0$ and $\beta_{12}=0$.


(1) {\it Case $\beta_{12}\ne0$.} By Theorem
\ref{trm:exist-hyp} in this case there exists one hyperbola if and
only if $\gamma_{15}=0$. We observe that because $b\ne0$ (since
$\mathcal{R}_{11}\ne0$) we may assume $b=1$ by the rescaling
$(x,y,t)\mapsto(bx,y,t/b)$. Since $3g-1\ne0$, setting
$c=(3g-1)c_1$ the condition $\gamma_{15}=0$ yields $a=c_1^2
(2g-1)$ and renaming the parameter $c_1$ as $c$ again we arrive
at the 2-parameter family of systems
\begin{equation} \label{sys:eta=theta=mu0=0-hyp-4}
 \frac {dx}{dt}= (c + x)\big[c(2 g-1) + g x\big],\quad
 \frac {dy}{dt}= 1 +(g-1)x y
\end{equation}
for which the condition $N\beta_{11}\beta_{12}\mathcal{R}_{11}$
implies
\begin{equation} \label{cond:4}
(g-1)(g+1)(2g-1)(3g-1)\ne0.
\end{equation}
 These systems posses the
following invariant hyperbola and invariant lines:
\begin{equation} \label{Hyp:4b}
\begin{gathered}
 \Phi(x,y)= \frac{1}{2g-1}+cy+ xy=0,\\
 L_1=gx+c(2 g-1) =0,\quad L_2=x+c  =0.
\end{gathered}
\end{equation}
On the other hand for systems \eqref{sys:eta=theta=mu0=0-hyp-4} we
calculate
\begin{equation} \label{val:mu0,mu1,mu2}
\mu_0=\mu_1=0,\quad \mu_2= c^2g (g-1)^2(2g-1) x^2,\quad
\gamma_{16}= c(g-1)^2 (1 - 3 g) x^3/2
\end{equation}
and by Lemma \ref{lem:mu_i-ISPs} in the case $\mu_2\ne0$ these
systems possess finite singular points of total multiplicity
two. Other two points have gone to infinity and coalesced with the
singularity $[0,1,0]$. So we consider two cases: $\mu_2\ne0$ and
$\mu_2=0$.


(a) {\it Subcase $\mu_2\ne0 $.} Then $c\ne0$ and
by the rescaling $(x,y,t)\mapsto(cx,y/c,t/c)$ we may assume
$c=1$. In this case the 1-parameter family of systems
\eqref{sys:eta=theta=mu0=0-hyp-4} possess the finite singular
points $M_i(x_i,y_i)$ (i=1,2) with the coordinates
$$
 x_1=\frac{(1 - 2
g)}{g},\quad y_1= \frac{g}{(g-1)(2g-1)},\quad x_2= -1,\quad y_2=
\frac{1}{g-1}.
$$
We detect that the singular point $M_1$ is located at the
intersection point of the hyperbola with invariant line $L_1=0$
(see \eqref{Hyp:4}) whereas $M_2$ is located on the line $L_2=0$
outside the hyperbola.

On the other hand taking into account \eqref{val:mu0,mu1,mu2} for
systems \eqref{sys:eta=theta=mu0=0-hyp-4} with $c=1$ we have
$\gamma_{16}\ne0 $ (because \eqref{cond:4}) and hence by Theorem
\ref{trm:exist-hyp} (see  Diagram \ref{diagr:eta=0}) the hyperbola
\eqref{Hyp:4} is a simple one. So considering the condition
\eqref{cond:4} and looking at all the intervals given by this condition
we arrive at the unique configuration presented by
Config. $\widetilde{{\rm H}}.19$.

(b) {\it Subcase $\mu_2=0 $.} Then considering
\eqref{val:mu0,mu1,mu2} and condition \eqref{cond:4} we get
$cg=0$ and we consider two possibilities: $\gamma_{16}\ne0 $ and
$\gamma_{16}=0$.


(b1) {\it Possibility $\gamma_{16}\ne0 $.} Then
$c\ne 0$ (and we may assume $c=1$) and this implies $g=0$. So we
arrive at the system with constant coefficients
$$
 \frac {dx}{dt}= -(1 + x),\quad  \frac {dy}{dt}= 1 -x y
$$
possessing one finite singular point $M_1(-1,-1)$, the invariant
hyperbola $
 x y+ y-1 =0 $ and the invariant line $x+1=0$.
On the other hand following Lemma \ref{lem:line-L} we detect that
the line at infinity $Z=0$ is double for these systems because $Z$
is a common factor of degree one of the polynomials
$\mathcal{E}_1(X,Y,Z)$ and ${\mathcal E}_2(X,Y,Z)$. Moreover, since
$\mu_0=\mu_1=\mu_2=0$ and $\mu_3=-x^2y$, according to Lemma
\ref{lem:mu_i-ISPs} we deduce that another finite singular point
has gone to infinity and coalesced with $[1,0,0]$. We observe that
$M_1$ belongs to the invariant line and it is outside the
hyperbola, i.e. we get Config. $\widetilde{{\rm H}}.24$.

(b2) {\it Subcase $\gamma_{16}=0 $.} In this case
$c=0$ and we get the systems
$$
 \frac {dx}{dt}= g x^2,\quad  \frac {dy}{dt}= 1 +(g-1)x y,
$$
for which $g\ne0$ (otherwise we obtain a degenerate system). For
these systems we calculate
$$
\mu_0=\mu_1=\mu_2=\mu_3=\gamma_{16}=0,\quad \mu_4=g^2x^4,\quad
\delta_6=(g-1) (4g-1) (x^2)/2
$$
and by Lemma \ref{lem:mu_i-ISPs} we deduce that all four finite
singular points have gone to infinity and coalesced with
$[0,1,0]$. Moreover, for the above systems we calculate
$$
\mathscr{E}_k(X)=g x^3 (1 + g - x y + g x y)
$$
and by Lemma
\ref{lem:Ek} the invariant line $x=0$ is a triple one.

According to  Diagram \ref{diag:eta=0-chi} the hyperbola is simple if $\delta_6\ne0$
(i.e. $4g-1\ne0$) and it is double if $\delta_6=0$ (i.e.
$4g-1=0$). So we arrive at Config. $\widetilde{{\rm H}}.25$ if
$\delta_6\ne0$ and at Config. $\widetilde{{\rm H}}.34$ if
$\delta_6=0$.


(2) {\it Case $\beta_{12}=0$.} Then $g=1/3$ and we
calculate $\gamma_{16}=- 2 c x^3 /9 $. Since by Theorem
\ref{trm:exist-hyp} in the case under consideration the condition
$\gamma_{16}=0$ is necessary for the existence of an invariant
hyperbola, we obtain $c=0$ and we arrive at the 1-parameter family of
systems
$$
 \frac {dx}{dt}= a + x^2/3,\quad  \frac {dy}{dt}= 1 - 2 x y/3.
$$
For these systems we calculate $\gamma_{17}=32 a x^2/9$ and
following Theorem \ref{trm:exist-hyp} we conclude that for
$\gamma_{17}<0$ or $\gamma_{17}>0$ or $\gamma_{17}=0$ we obtain
three different configurations by the number and types of
hyperbolas. Since $\operatorname{sign}(a)=\operatorname{sign}(\gamma_{17})$ setting a new
parameter $k$ as follows: $a= \operatorname{sign}(a)k^2/3$ after the rescaling
$(x,y,t)\mapsto(kx,3y/k, 3t/k)$ (in the case $k\ne0$) or the
rescaling $x\to 3x$ if $a=0$, the above systems become
\begin{equation} \label{sys:eta=theta=mu0=0-hyp-5}
 \frac {dx}{dt}= x^2+\varepsilon,\quad  \frac {dy}{dt}=1 - 2 x y,
\end{equation}
where $\varepsilon=\operatorname{sign}(\gamma_{17})$ if $\gamma_{17}\ne0$ and
$\varepsilon=0$ if $\gamma_{17}=0$, i.e.
$\varepsilon\in\{-1,0,1\}$.

These systems possess the following invariant hyperbolas and
invariant lines:
\begin{equation} \label{Hyp:5}
 \Phi_{1,2}(x,y)= 3 \pm\sqrt{-\varepsilon} y - x y=0,\quad
L_{1,2}=x \pm\sqrt{-\varepsilon} =0.
\end{equation}
We detect that these systems possess the finite singularities
$M_{1,2}\big(\pm\sqrt{\varepsilon},
3\pm1/(2\sqrt{\varepsilon})\big)$ (if $\varepsilon\ne0$) and each
one of the lines intersect only one of the hyperbolas.

On the other hand for systems
\eqref{sys:eta=theta=mu0=0-hyp-5} we calculate
$$
\mu_0=\mu_1=0, \quad \mu_2= 4 \varepsilon x^2,\quad \mu_3=0,
\quad \mu_4= x^2 (x + 2\varepsilon y)^2.
$$
Therefore by Lemma \ref{lem:mu_i-ISPs} we conclude that in the
case $\varepsilon\ne0$ only two finite singularities of these
systems have gone to infinity and coalesced with $[0,1,0]$ and we
get Config. $\widetilde{{\rm H}}.27$ if $\gamma_{17}<0$ and
Config. $\widetilde{{\rm H}}.28$ if $\gamma_{17}>0$.

Assume now $\gamma_{17}=0$ (i.e. $\varepsilon=0$). Then
$\mu_i=0$ for $i=0,1,2,3$ and $\mu_4= x^4$ and by Lemma
\ref{lem:mu_i-ISPs} all the finite singularities of this system
have gone to infinity and coalesced with $[0,1,0]$.

We observe that the two lines coincide and we get the invariant
multiple line $x=0$. Considering Lemma \ref{lem:Ek} for systems
\eqref{sys:eta=theta=mu0=0-hyp-5} with $\varepsilon=0$ we calculate
$$
\mathscr{E}_k(X)= 2 x^3 (2 - 3 x y)
$$
and by this lemma in the case under consideration the
invariant line $x=0$ is a triple one.
 Since by Theorem \ref{trm:exist-hyp} (see  Diagram
\ref{diagr:eta=0}) the hyperbola
\eqref{Hyp:5} in the case $\gamma_{17}=0$ (i.e. $\varepsilon=0$)
is double, we arrive at the same configuration given by
Config. $\widetilde{{\rm H}}.34$.


\subparagraph{Possibility $N=0$.} Then $(g-1)(g+1)=0$ and as
$\beta_{13}=(g-1)^2 x^2/4$ we consider two cases: $\beta_{13}\ne0$
and $\beta_{13}=0$.


(1) {\it Case $\beta_{13}\ne0$.} Therefore the
condition $N=0$ gives $g=-1$ and we can assume $e=f=0$ by a
translation. So we get the systems
$$
 \frac {dx}{dt}= a + c x + d y - x^2,\quad
 \frac {dy}{dt}=b - 2 x y,
$$
which by Theorem \ref{trm:exist-hyp} (see Diagram
\ref{diagr:eta=0}) possess an invariant hyperbola if and only if
$\gamma_{10}=\gamma_{17}=0$ and $\mathcal{R}_{11}\ne0$.
Calculations yield
\begin{gather*}
\gamma_{10}=14 d^2=0,\quad \gamma_{17} = -8 (16 a + 3 c^2) x^2 + 4
d y (14 c x + 9 d y)=0,\\ \mathcal{R}_{11}=-6 x (2 b x^3 - c d x
y^2 - d^2 y^3)\ne0
\end{gather*}
and therefore we obtain $d=0$, $a=- 3 c^2 /16$ and $b\ne0$ and we
may assume $b=1$ by the rescaling $y\to by$. So we arrive at
the 1-parameter systems
$$
 \frac {dx}{dt}= - 3 c^2/16+ c x - x^2,\quad  \frac {dy}{dt}=1 - 2 x y
$$
possessing the invariant hyperbola and the invariant lines
\begin{equation} \label{Hyp:6b2}
 \Phi(x,y)= 4 + 3c y -12 x y=0,\quad L_1=4x -c =0,\quad L_2=4x -3c=0.
\end{equation}
We observe that for $c=0$ the lines coincide and this phenomenon
is governed by the invariant polynomial $\gamma_{16}=-2 c x^3$.
So we consider two subcases: $\gamma_{16}\ne0$ and
$\gamma_{16}=0$.


(a) {\it Subcase $\gamma_{16}\ne0$.} Then $c\ne0$
and we may assume $c=4$ by the rescaling
$(x,y,t)\mapsto(cx/4, 4y/c, 4t/c)$. So we obtain the system
 \begin{equation} \label{sys:eta=theta=mu0=0-hyp-6a}
 \frac {dx}{dt}= (x-1) (3-x),\quad  \frac {dy}{dt}=1 - 2 x y
\end{equation}
which possesses the following invariant hyperbola and invariant
lines:
\begin{equation} \label{Hyp:6a2}
 \Phi(x,y)= 1/3 + y - x y=0,\quad L_1=x -1 =0,\quad L_2=x -3=0
\end{equation}
and two finite singularities: $M_1(1,1/2)$ and $M_2(3,1/6)$. Since
$\mu_0=\mu_1=0$ and $\mu_2=12 x^2$ by Lemma~\ref{lem:mu_i-ISPs}
we conclude that two finite singularities of this system have
gone to infinity and coalesced with $[0,1,0]$. So considering the
position of the hyperbola, invariant lines and of the finite
singularities we arrive at Config. $\widetilde{{\rm H}}.19$.


(b) {\it Subcase $\gamma_{16}=0$.} Then $c=0$ and
we obtain the system
$$
 \frac {dx}{dt}= - x^2,\quad  \frac {dy}{dt}=1 - 2 x y,
$$
for which
$$
\mu_0=\mu_1=\mu_2=\mu_3=0, \quad \mu_4= x^4.
$$
So by Lemma \ref{lem:mu_i-ISPs} all the finite singularities of
this system have gone to infinity and coalesced with $[0,1,0]$.

On the other hand we observe that the invariant line $x=0$ is a multiple one.
For the above system we calculate
$\mathscr{E}_k(X)=2 x^4 y$ and by Lemma \ref{lem:Ek} we deduce
that the invariant line $x=0$ has multiplicity four. So
considering the invariant hyperbola \eqref{Hyp:6a} (for $c=0$) we
arrive at the configuration given by Config. $\widetilde{{\rm H}}.26$.


(2) {\it Case $\beta_{13}=0$.} Then we have $g=1$ and
we can assume $c=0$ by a translation. So we get the systems
$$
 \frac {dx}{dt}= a + d y+ x^2,\quad  \frac {dy}{dt}=b + e x + f y,
$$
and by Theorem \ref{trm:exist-hyp} (see  Diagram
\ref{diagr:eta=0}) these systems possess an invariant hyperbola if
and only if $\gamma_9=\tilde \gamma_{18}=\tilde \gamma_{19}=0$.
Calculations yield
$$
\gamma_9=-6 d^2=0,\quad \tilde \gamma_{18}=8 x^2 (e x^2 - 2 d
y^2)=0,\quad \tilde \gamma_{19}=4 (4 a + f^2) x + 4 d f y=0
$$
and evidently this implies $d=e=0$ and $a=-f^2/4$ which leads to
the 2-parameter family of systems
$$
 \frac {dx}{dt}= - f^2/4 + x^2,\quad  \frac {dy}{dt}=b + f y.
$$
For these systems we calculate $\mu_0=\mu_1=0$, $\mu_2=f^2 x^2$
and we consider two subcases: $\mu_2\ne0$ and $\mu_2=0$.


(a) {\it Subcase $\mu_2\ne0$.} Then $f\ne0$ and we
may assume $f=1$ and $b=0$ because of the transformation
$(x,y,t)\mapsto(fx, y-b/f, t/f)$. So we obtain the system
 \begin{equation} \label{sys:eta=theta=mu0=0-hyp-7}
 \frac {dx}{dt}= (2x-1) (2x+1)/4,\quad  \frac {dy}{dt}=y
\end{equation}
which possesses the 1-parameter family of hyperbola:
$$
 \Phi(x,y)= -q/2 + q x + y + 2 x y=0,\quad q\in \mathbb{C}\setminus\{0\}
$$
as for $q=0$ we get a reducible conic.

On the other hand system \eqref{sys:eta=theta=mu0=0-hyp-7}
possesses the following invariant lines and finite singularities:
$$
L_1=2x -1 =0,\quad L_2=2x +1=0,\quad L_3=y=0,\quad M_{1,2}(\pm
1/2,0).
$$
Following Lemmas \ref{lem:line-L} and \ref{lem:Ek} for this system
we calculate
\begin{gather*}
\gcd\big({\mathcal E}_1(X,Y,Z),{\mathcal E}_2(X,Y,Z)\big)=YZ(2 X -
Z)^2 (2 X + Z),\\
\mathscr{E}_k(X)=(1 - 2 x)^2 (1 + 2 x) y/4
\end{gather*}
and we deduce that the invariant lines $ L_2=0$ and $ L_3=0$ are
simple, whereas the line $L_1=0$ as well as the infinite line
$Z=0$ are double ones.

So considering the fact that other two finite singular points have
gone to infinity and coalesced with $[1,0,0]$ we arrive at
Config. $\widetilde{{\rm H}}.35$.


(b) {\it Subcase $\mu_2=0$.} In this case we have
$f=0$ and as $b\ne0$ (otherwise we get degenerate system) we may
assume $b=1$ by the change $y\to by$ and we get the system
\begin{equation} \label{sys:eta=theta=mu0=0-hyp-8}
 \frac {dx}{dt}= x^2,\quad  \frac {dy}{dt}=1
\end{equation}
which possesses the 1-parameter family of hyperbola:
$$
 \Phi(x,y)= 1 + r x + x y=0,\quad r\in \mathbb{C}
$$
and has no finite singularities. Calculations yield
\begin{gather*}
\mu_0=\mu_1=\mu_2=\mu_3=0, \quad \mu_4= x^4,\\
\gcd\big({\mathcal E}_1(X,Y,Z),{\mathcal E}_2(X,Y,Z)\big)=X^3
Z^2,\quad \mathscr{E}_1(X)=2 X^3
\end{gather*}
and considering Lemma \ref{lem:mu_i-ISPs} we conclude that all
the finite singularities of these systems have gone to infinity
and coalesced with $[0,1,0]$. Moreover by Lemmas \ref{lem:line-L}
and \ref{lem:Ek} the invariant line $x=0$ as well as the infinite
line $Z=0$ are of multiplicity 3. As a result we arrive at the
configuration given by Config. $\widetilde{{\rm H}}.36$.

\subsection{Possibility $M(\tilde a,x,y)=0=C_2(\tilde a,x,y)$}

In this section we consider the configurations
of invariant hyperbolas and invariant lines of quadratic systems
with $C_2=0$, taking into account Theorem \ref{trm:exist-hyp}
(see  Diagram \ref{diagr:eta=0}). Then the line at infinity is filled up with
singularities and according to \cite{SchVul08-JDDE} in this case
via an affine transformation and time rescaling quadratic
systems could be brought to the following systems
\begin{equation} \label{sys:C2=0-Gen}
 \dot x=k+cx+dy+x^2,\quad \dot y=l + xy.
\end{equation}
Following \cite{SchVul08-JDDE} we consider the stratification of
the parameter space of the above systems given by invariant
polynomials $H_9-H_{12}$ in \cite[Table 1 on page
754]{SchVul08-JDDE} according to possible configurations of
invariant lines. So for systems \eqref{sys:C2=0-Gen} we calculate
$H_{10}=36 d^2$ and we consider two cases: $H_{10}\ne0$ and
$H_{10}=0$.

\subsubsection{Case $H_{10}\ne0$}
Then $d\ne0$ and as it was shown in \cite[pages 748,749]{SchVul08-JDDE},
in this case via some parametrization and
using an additional affine transformation and time rescaling we
arrive at the following 2-parameter family of systems
\begin{equation} \label{sys:C2=0-d-ne0}
 \dot x=a+y+ (x+c)^2,\quad \dot y= xy.
\end{equation}
for which we calculate
$$
 N_7 =16 c (9 a + c^2),\quad H_9=2304 a (a + c^2)^2
$$
and by Theorem \ref{trm:exist-hyp} (see  Diagram \ref{diagr:eta=0})
for the existence of invariant hyperbola the condition $N_7=0$ is
necessary and sufficient. So we have either $c=0$ or $9 a +
c^2=0$. However in the second case the condition $a\le0$ must hold
and in the case $a=0$ we get again $c=0$. In the case $a<0$ we may
assume $a=-1$ and $c>0$ by the rescaling
$\big(x,y,t)\mapsto(\operatorname{sign}(c)\sqrt{-a}\,x, -ay,
t/(\operatorname{sign}(c)\sqrt{-a})\big)$, therefore we set $c=3$. Moreover the transformation
$$
(x,y,t)\mapsto\big(2(x-1),\ 4(y-x-1),\ t/2\big).
$$
sends the system \eqref{sys:C2=0-d-ne0} for $a=-1$, $c=3$ to the system
\eqref{sys:C2=0-d-ne0} with $a=-1$ and $c=0$.
Thus we assume $c=0$ and we get the systems
\begin{equation} \label{sys:C2=0-d-ne0-c=0-Hyp}
 \dot x=a+y+ x^2,\quad \dot y= xy
\end{equation}
which possess the following 1-parameter family of hyperbolas
\begin{equation} \label{Hyp:7-a}
 \Phi(s,x,y)=a + 2 y + x^2 - m^2 y^2=0
\end{equation}
as well as the following invariant lines and finite singularities:
$$
L_1=y=0,\quad L_{2,3}=a x^2+(a + y)^2=0;\quad M_1(0,-a),\quad
M_{2,3}(\pm\sqrt{-a},\,0).
$$
We observe that the two lines $L_{2,3}=0$ as well as the
singular points $M_{2,3}$ are real if $a<0$; they are complex if
$a>0$ and they coincide if $a=0$. Moreover these three
possibilities are distinguished by the invariant polynomial
$H_9=2304 a^3$.

So, considering that all the hyperbolas from the family
\eqref{Hyp:7-a} intersect the invariant line $y=0$ at the singular
points $M_{2,3}$ we arrive at the configuration
Config. $\widetilde{{\rm H}}.39$ if $H_9<0$;
Config. $\widetilde{{\rm H}}.41$ if $H_9>0$ and
Config. $\widetilde{{\rm H}}.43$ if $H_9=0$.

\subsubsection{Case $H_{10}=0$} In this case we have $d=0$ and
we distinguish two subcases: $k\ne0$ and $k=0$. Since for
systems \eqref{sys:C2=0-Gen} with $d=0$ we have $H_{12}=-8 k^2
x^2$ it is clear that this invariant polynomial governs these two
subcases.

\paragraph{Subcase $H_{12}\ne0$.} Then $k\ne0$ and as it
was shown in \cite[page 750]{SchVul08-JDDE} in this case via an
affine transformation and time rescaling after some additional
parametrization we arrive at the following 2-parameter family of
systems
\begin{equation} \label{sys:C2=0-d=0-k-ne0}
 \dot x=a+ (x+c)^2,\quad \dot y= xy.
\end{equation}
For these systems the condition $H_{12}=-8 (a + c^2)^2 x^2\ne0$
must hold and according to  Diagram \ref{diagr:eta=0} the condition
$N_7 =16 c (9 a + c^2)=0$ must be satisfied for the existence
of invariant hyperbolas. On the other hand for these systems we
have $H_2=8 c x^2$ and we consider two possibilities: $H_2=0$ and
$H_2\ne0$.

\subparagraph{Possibility $H_2\ne0$.} Then $c\ne0$ and in
this case we get $9a+c^2=0$, i.e. $a=-c^2/9\ne0$. Therefore by
the rescaling $(x,y,t)\mapsto(2cx, y, t/(2c))$ systems
\eqref{sys:C2=0-d=0-k-ne0} could be brought to the system
\begin{equation} \label{sys:C2=0-d=0-k-ne0-Hyp-2-a}
 \dot x=(1 + 3 x) (2 + 3 x)/9,\quad \dot y= xy.
\end{equation}
This system possesses the 1-parameter family of the hyperbolas and
three invariant lines
\begin{equation} \label{Hyp:8-a}
\Phi(x,y)= 4 + 12 x + 9 x^2 + m y + 3 m x y=0;\quad y=0,\quad
3x+1=0,\quad 3x+2=0,
\end{equation}
as well as the singularities $M_1(-1/3,0)$ and $M_2(-2/3,0)$. It
is not too difficult to convince ourselves that in this case we
get the configuration given by Config. $\widetilde{{\rm H}}.37$.

\subparagraph{Possibility $H_2=0$.} Then $c=0$ and we get the
systems
\begin{equation} \label{sys:C2=0-d=0-k-ne0-Hyp-1-a}
 \dot x=a + x^2,\quad \dot y= xy,\quad a\ne0,
\end{equation}
which possess the following family of conics and the invariant
lines:
\begin{equation} \label{Hyp:9-a}
\Phi(x,y)= a + x^2 -m^2 y^2=0;\quad L_1=y=0,\quad L_{2,3}=x^2+a=0
\end{equation}
as well as two finite singularities: $M_{1,2}(\pm\sqrt{-a},\,y)$.

On the other hand we calculate $H_{11}=-192 a x^4$; therefore
$\operatorname{sign}(a)=-\operatorname{sign}(H_{11})$. So conside\-ring the position of the
invariant lines and of the hyperbolas given in
\eqref{Hyp:9-a} we obtain the configuration
Config. $\widetilde{{\rm H}}.40$ if $H_{11}<0$ and
Config. $\widetilde{{\rm H}}.38$ if $H_{11}>0$.


\paragraph{Subcase $H_{12}=0$.} Then $k=0$ and we arrive at
the family of systems \eqref{sys:C2=0-Gen} with $d=k=0$ for which
we have $N_7=-16 c^3$ and by Theorem \ref{trm:exist-hyp} (see
 Diagram \ref{diagr:eta=0}) we have to force the condition $c=0$.
Since $l\ne0$ (otherwise we get a degenerate system) by the
change $y\to ly$ we may assume $l=1$ and we arrive at the system
\begin{equation} \label{sys:C2=0-d=0-k=0-Hyp-1-a}
 \dot x= x^2,\quad \dot y=1 + xy,
\end{equation}
which possesses the following family of hyperbolas
$$
\Phi(x,y)= 1 + m x^2 + 2 x y=0
$$
and the invariant line $x=0$. We remark that by Lemma \ref{lem:Ek}
this line is triple since for this system we have
$\mathscr{E}_1(X)= X^3$. So considering the absence of finite
singularities of system
\eqref{sys:C2=0-d=0-k=0-Hyp-1-a} we obtain the configuration given
by Config. $\widetilde{{\rm H}}.42$.

This completes the proof of statement $(B)$ of Main Theorem.

\section{Concluding comments}\label{sec:conclusion}

Details about the configurations and their realizability.
Diagrams~\ref{diag:eta>0,theta-ne0},
\ref{diag:eta>0,theta=0} and \ref{diag:eta=0-chi}
give an algorithm to compute the
configuration of a system with an invariant hyperbola for any
system presented in any normal form and they are also the
bifurcation diagrams of the configurations of such systems, done
in the 12-parameter space of the coefficients of these systems.

\subsection{Concluding comments for $\eta>0$}

In this section we consider the class of all non-degenerate
systems in QSH$_{(\eta>0)}$. According to Theorem \ref{th:Main1}, this class
yields 162 distinct configurations which can be split according the
following geometric classification.

(A1) There are exactly 3 configurations of systems possessing an infinite
number of hyperbolas, namely Config. H.160, Config. H.161 and Config. H.162,
 which are distinguished by the number and multiplicity of the invariant
straight lines of such systems.

(A2) The remaining 159 configurations could have up to a maximum of 3
distinct invariant hyperbolas, real or complex, and up to 4 distinct
invariant straight lines, real or complex, including the line at infinity.
 Assuming we have $m$ invariant hyperbolas $H_i :f_i(x,y)=0$ and $m'$
invariant lines $L_j :g_j(x,y)=0$, the geometry of the configurations is
in part captured by the following invariants:
 \begin{enumerate}
\item[(a)] the type of the main divisor $\sum n(H_i)H_i+\sum n(L_j)L_j$
 on the plane $P_2(\mathbb{R})$, where $n(H_i)$, $n(L_j)$ indicate the multiplicity
 of the respective invariant curve;

\item[(b)] the type of the zero-cycle $MS_{0C}=\sum l_iU_i+\sum m_js_j$ on the
 plane $P_2(\mathbb{R})$, where $l_i$, $m_j$ indicate the multiplicity on the real
 projective plane, of the real singularities at infinity $U_i$ and in the finite
 plane $s_j$ of a system \eqref{sys:QSgenCoef}, located on the invariant lines
 and invariant hyperbolas;

\item[(c)] the number of the singular points of the systems which are smooth
points of the curve: $T(X,Y,Z)=\prod F_i(X,Y,Z)\cdot \prod G_j(X,Y,Z)\cdot Z=0$
 where $F_i,G_j$'s are the homogenizations of $f_i$'s, $g_j$'s respectively,
 where $f_i=0$ are the invariant hyperbolas and $g_j=0$ are the invariant
straight lines, and by their positions on $T(X,Y,Z)=0$. This position is
expressed in the proximity divisor PD on the Poincar\'e disk of a system,
 defined in Section \ref{sec:auxil results}.
 \end{enumerate}

 We have exactly 120 distinct configurations of systems with exactly one
hyperbola which is simple:
 \begin{itemize}
 \item[(i)] 40 of them with no invariant line other than the line at infinity:
 36 of them having only a simple line at infinity, 2 of them having a double
 line at infinity, and 2 of them having a triple line at infinity;

 \item[(ii)] 46 of them with only one invariant line other than the line at infinity:
 39 of them having only simple lines, 3 of them with a double finite line, and
4 of them with the line at infinity being double;

 \item[(iii)] 23 of them with two distinct simple affine invariant lines
 (real or complex)  and a simple line at infinity;
 \item[(iv)] 6 of them with three simple invariant straight lines other than
the line at infinity;

 \item[(v)] 2 of them with two simple lines and one double line: 1 of them with
 a double finite line and 1 of them with a double line at infinity;

 \item[(vi)] 3 of them with four simple invariant straight lines other than
the line at infinity.
 \end{itemize}

We have exactly 35 distinct configurations with hyperbolas of total multiplicity
two:
 \begin{itemize}
 \item[(vii)] 11 of them with no invariant straight line other than the line
at infinity;
 \item[(viii)] 5 of them with only one simple invariant straight line other
than a simple line at infinity;
 \item[(ix)] 11 of them with exactly two invariant lines which are simple
other than the line at infinity, which 2 of them with a double hyperbola;
 \item[(x)] 3 of them with exactly one double line either in the finite plane
 or at infinity;
 \item[(xi)] 5 of them with three simple invariant straight lines other than
the line at infinity.
 \end{itemize}

 We have exactly 4 distinct configurations with three distinct hyperbolas:
 \begin{itemize}
 \item[(xii)] 2 of them with only one invariant straight line other than the
line at infinity;
 \item[(xiii)] 2 of them with exactly two invariant lines which are simple
other than the line at infinity.
 \end{itemize}

\subsection{Concluding comments for $\eta=0$}

In this section we consider the class QSH$_{(\eta=0)}$ of all non-degenerate
quadratic differential systems \eqref{sys:QSgenCoef} possessing an invariant
 hyperbola and either exactly two distinct real
singularities at infinity or the line at infinity filled up with
singularities. According to Theorem \ref{th:Main2}, this class
yields 43 distinct configurations which can be split according the
following geometric classification.

(A1) There are exactly 9 configurations with an infinity of invariant hyperbolas.
These configurations could have up to 3 distinct affine invariant lines which
could have multiplicities up to at most 3. The configurations are split as follows:
 \begin{itemize}
 \item[(a)] 2 of them with exactly two infinite singularities
(Config. $\widetilde{{\rm H}}.35$ and Config. $\widetilde{{\rm H}}.36$)
distinguished by the type of the invariant lines divisor ILD
(as defined in Section \ref{sec:auxil results});
 \item[(b)] 7 of them with the line at infinity filled up with singularities
(Config. $\widetilde{{\rm H}}.i$, with $37\leq i\leq 43$).
The type of the ILD splits these configurations in three groups:\\
 \emph{Group 1:} Config. $\widetilde{{\rm H}}.i$, with
$37\leq i\leq 39$, first distinguished by the number of finite singularities
(3 for Config. $\widetilde{{\rm H}}.39$ and 2 for Config. $\widetilde{{\rm H}}.i$,
$i\in \{37,38\})$. The last two configurations are distinguished by the number
of finite singularities not located on the invariant hyperbolas
(1 for $i=37$, 0 for $i=38$);\\
\emph{Group 2:} Config. $\widetilde{{\rm H}}.i$, with $i\in \{40,41\}$; and\\
\emph{Group 3:} Config. $\widetilde{{\rm H}}.i$, with $i\in \{42,43\}$.
The configurations in these groups are distinguished by the type of the zero-cycle
 MS$_{0C}$;
 \end{itemize}

(A2) The remaining 34 configurations could have up to a maximum of 2 distinct
invariant  hyperbolas, real or complex, and up to 3 distinct invariant straight
 lines, real or  complex, including the line at infinity.

 We have exactly 11 distinct configurations of systems with exactly one
hyperbola which is simple, and no  invariant affine lines. These are classified
by the total multiplicity of the real  singularities of the systems located on
the algebraic solutions ($TMS$) as follows:
 \begin{itemize}
 \item[(a)] only one configuration (Config. $\widetilde{{\rm H}}.1$) with $TMS=3$;

 \item[(b)] 5 configurations with $TMS=5$ grouped as follows by the number of
their singularities and their multiplicities:
 \subitem -- one with only two singularities, both multiple and both at
 infinity (Config. $\widetilde{{\rm H}}.2$);
 \subitem -- two with an additional finite singularity (Config.
$\widetilde{{\rm H}}.3$, Config. $\widetilde{{\rm H}}.4$) but with distinct
 multiplicities;
 \subitem -- two with two additional finite simple singularities
(Config. $\widetilde{{\rm H}}.5$, Config. $\widetilde{{\rm H}}.6$)
distinguished using the proximity divisor
 PD defined in Section \ref{sec:auxil results};

 \item[(c)] 4 with $TMS=6$: one with only one finite singularity
(Config. $\widetilde{{\rm H}}.7$);\ 3 with two finite singularities
with the same multiplicities, distinguished by the
 invariant $O$ defined in Section \ref{sec:auxil results}
(Config. $\widetilde{{\rm H}}.i$, with $8\leq i\leq 10$);

 \item[(d)] 1 with $TMS=7$ (Config. $\widetilde{{\rm H}}.11$).
 \end{itemize}

 We have exactly 6 distinct configurations with a unique simple invariant
hyperbola and a unique simple invariant line:
 \begin{itemize}
 \item[(e)] one with no finite singularity (Config. $\widetilde{{\rm H}}.12$);
 \item[(f)] one with only one finite singularity located on the hyperbola
(Config. $\widetilde{{\rm H}}.13$);
 \item[(g)] two with three finite singularities
(Config. $\widetilde{{\rm H}}.13$, Config. $\widetilde{{\rm H}}.15$),
distinguished by the number of finite singularities located on the invariant line;
 \item[(h)] two with four simple finite singularities
 (Config. $\widetilde{{\rm H}}.16$, Config. $\widetilde{{\rm H}}.17$),
 which are distinguished by the proximity divisor PD
(see Section \ref{sec:auxil results});
 \end{itemize}

We have exactly 9 distinct configurations with a simple invariant hyperbola
and invariant lines, including the line at infinity, of total multiplicity
$3\le TML \le5$:
 \begin{itemize}
 \item[(i)] 5 configurations have exactly three distinct simple invariant
lines (Config. $\widetilde{{\rm H}}.i$, $18\leq i\leq 22$) distinguished
by the types of $ICD, MS_{0C}$ and the proximity divisor PD;
 \item[(j)] 4 configurations with exactly two invariant lines, one of
them being multiple (Config. $\widetilde{{\rm H}}.17$, $23\leq i\leq 26$).
They are distinguished by the multiplicities of the two invariant lines.
 \end{itemize}

We have exactly 8 distinct configurations with invariant hyperbolas of total
multiplicity 2:
 \begin{itemize}
 \item[(k)] two with two distinct hyperbolas, one with real hyperbolas (Config. $\widetilde{{\rm H}}.27$) and one with complex (non-real) hyperbolas (Config. $\widetilde{{\rm H}}.28$),
 \item[(l)] six of them with a double hyperbola, one with 4 finite singularities (Config. $\widetilde{{\rm H}}.32$), one with 3 finite singularities (Config. $\widetilde{{\rm H}}.33$), one with 2 finite singularities (Config. $\widetilde{{\rm H}}.30$), and three without any finite singularity (Config. $\widetilde{{\rm H}}.29$, Config. $\widetilde{{\rm H}}.31$, Config. $\widetilde{{\rm H}}.34$), distinguished by the presence and multiplicity of the finite invariant line;
 \end{itemize}


\subsection*{Acknowledgments}
The authors thank the anonymous referee for useful comments and minor corrections.
R. D. S. Oliveira was  supported by CNPq grant ``Projeto Universal''
 472796/2013-5, by
CAPES CSF-PVE-88881.030454/2013-01 and by Projeto Tem\'atico
FAPESP number 2014/00304-2.
R. D. S. Oliveira and N. Vulpe were supported by FP7-PEOPLE-2012-IRSES-316338.
A. C. Rezende was supported by CNPq-PDE 232336/2014-8 grant.
D. Schlomiuk and N. Vulpe were  supported by the NSERC Grant RN000355.
 N. Vulpe  was supported by the project
15.817.02.03F from SCSTD of ASM.


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\end{document}