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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 159, pp. 1--79.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/159\hfil Geometric configurations of singularities]
{Geometric configurations of singularities for quadratic
differential systems with \\ total finite multiplicity $m_f=2$}

\author[J. C. Art\'es, J. Llibre, D. Schlomiuk, N. Vulpe \hfil EJDE-2014/159\hfilneg]
{Joan C. Art\'es, Jaume Llibre, Dana Schlomiuk, Nicolae Vulpe}  % in alphabetical order

\address{Joan C. Art\'es \newline
Departament de Matem\`atiques, Universitat Aut\`onoma
de Barcelona, 08193 Bellaterra, Barcelona, Spain}
\email{artes@mat.uab.cat}

\address{Jaume Llibre \newline
Departament de Matem\`atiques, Universitat Aut\`onoma
de Barcelona, 08193 Bellaterra, Barcelona, Spain}
\email{jllibre@mat.uab.cat}

\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 Science of Moldova, 5 Academiei str, Chi\c{s}in\u{a}u,
MD-2028, Moldova}
\email{nvulpe@gmail.com}

\thanks{Submitted November 25, 2013. Published July 18, 2014.}
\subjclass[2000]{58K45, 34C05, 34A34}
\keywords{Quadratic vector fields;  infinite and finite singularities;
\hfill\break\indent affine invariant polynomials; Poincar\'e compactification;
 configuration of singularities;
\hfill\break\indent geometric equivalence relation}

\begin{abstract}
In this work we consider the problem of classifying all
configurations of singularities, both finite and infinite of
quadratic differential systems, with respect to the
\textit{geometric equivalence relation} defined in
\cite{Art-Llib-Schl-Vlp-RMJM-2013}. This relation is deeper than
the \textit{topological equivalence relation} which does not
distinguish between a focus and a node or between a strong and a
weak focus or between foci of different orders. Such distinctions
are however important in the production of limit cycles close to
the foci in perturbations of the systems. The notion of
\textit{geometric equivalence relation} of configurations of
singularities allows to incorporates all these important geometric
features which can be expressed in purely algebraic terms. This
equivalence relation is also deeper than the \textit{qualitative
equivalence relation} introduced in \cite{J_L}. The
\textit{geometric classification} of all configurations of
singularities, finite and infinite, of quadratic systems was
initiated in \cite{Art-Llib-Schl-Vlp-RMJM-CRM Report} where the
classification was done for systems with total multiplicity $m_f$
of finite singularities less than or equal to one.  In this
article we continue the work initiated in
\cite{Art-Llib-Schl-Vlp-RMJM-CRM Report} and obtain the
\textit{geometric classification} of singularities, finite and
infinite, for the subclass of quadratic differential systems
possessing finite singularities of total multiplicity $m_f=2$. We
obtain 197 \textit{geometrically distinct} configurations of
singularities for this family. We also give here the global
bifurcation diagram of configurations of singularities, both
finite and infinite, with respect to the \textit{geometric
equivalence relation}, for this class of systems.  The bifurcation
set of this diagram is algebraic. The bifurcation diagram is done
in the 12-dimensional space of parameters and it is expressed in
terms of polynomial invariants. The results can therefore be
applied for any family of quadratic systems in this class, given
in any normal form. Determining the geometric configurations of
singularities for any such family, becomes thus a simple task
using computer algebra calculations.

\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\tableofcontents

\renewcommand*{\tablename}{Diagram}

\section{Introduction and statement of main results}

We consider here  differential systems of the form
\begin{equation} \label{il1}
  \frac {dx}{dt}= p(x,y),\quad
  \frac {dy}{dt}= q(x,y),
\end{equation}
where $p,q\in \mathbb{R}[x,y]$, i.e. $p, q$ are polynomials in $x,y$ over $\mathbb{R}$.
We call \textit{degree} of a system \eqref{il1}
the integer $m=\max(\deg p,\, \deg q)$.
In particular we call \textit{quadratic} a differential
system \eqref{il1} with $m=2$.  We denote here by \text{QS} the
whole class of real  quadratic differential systems.

The study of the class \text{QS} has proved to be quite a
challenge since hard problems formulated more than a century ago,
are still open for this class. It is expected that we have a
finite number of phase portraits in \text{QS}.  Although we have
phase portraits for several subclasses of \text{QS}, the
complete list of phase portraits of this class is not known and
attempting to topologically classify these systems, which occur
rather often in applications, is a very complex task. This is
partly due to the elusive nature of limit cycles and partly to the
rather large number of parameters involved. This family of systems
depends on twelve parameters but due to the  group action of real
affine transformations and time homotheties, the class ultimately
depends on five parameters which is still a rather large number of
parameters. For the moment only subclasses depending on at most
three parameters were studied globally,   including global
bifurcation diagrams (for example \cite{Art-Llib-Schl-IJBCh}). On
the other hand we can restrict the study of the whole quadratic
class by focusing on specific global features of the systems in
this family. We may thus focus on the global study of
singularities and their bifurcation diagram. The singularities are
of two kinds: finite and infinite. The infinite singularities are
obtained by compactifying the differential systems on the sphere
or on the Poincar\'e disk as defined in Section \ref{sec:compact}
(see also \cite{Dum-Llib-Art}).

The global study of quadratic vector fields in the neighborhood of
infinity was initiated by Coll in \cite{Co:1987} where he
characterized all the possible phase portraits in a neighborhood
of infinity. Later on Nikolaev and Vulpe in \cite{Nikolaev-Vulpe}
classified topologically the singularities at infinity in terms of
invariant polynomials. Schlomiuk and Vulpe used geometric
concepts defined in \cite{Schlomiuk-Pal}, and also introduced some
new geometric concepts in \cite{Dana-Vlp-JDE} in order to
simplify the invariant polynomials and the classification. To
reduce the number of phase portraits in half, in both cases the
\textit{topological equivalence relation} was taken to mean the
existence of a homeomorphism of the plane carrying orbits to
orbits and \textit{preserving or reversing} the orientation. In
\cite{Art-Llib-Vlp-IJBCh} the authors classified topologically
(adding also the distinction between nodes and foci) the whole
quadratic class, according to configurations of their finite
singularities.

In the topological classification no distinction was made among the various
types of foci or saddles, strong or weak of various orders.  However
these distinctions of an algebraic nature are very important in the
study of perturbations of systems possessing such singularities.
Indeed, the maximum number of limit cycles which can be produced
close to the weak foci in perturbations depends on the orders of the
foci.

There are also three kinds of simple nodes as we can see in
Figure \ref{Nodes} below where the local phase portraits around
the singularities are given.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{fig1} % Nodes-1.eps
\caption{Different types of nodes.}\label{Nodes}
\end{center}
\end{figure}


In the three phase portraits of Figure \ref{Nodes} the
corresponding three singularities are stable nodes.  These portraits
are topologically equivalent but the solution curves do not arrive
at the nodes in the same way. In the first case, any two distinct
non-trivial phase curves arrive at the node with distinct slopes.
Such a node is  called a star node.  In the second picture all
non-trivial solution curves excepting two of them arrive at the node
with the same slope but the two exception curves arrive at the node
with a different slope. This is the generic node with two
directions. In the third phase portrait all phase curves arrive
at the node with the same slope. Here algebraic distinction means
that the linearization matrices at these nodes and their
eigenvalues, distinguish the nodes in Figure \ref{Nodes}, see
\cite{SchVul08-RMJM}.

We recall that the first and the third  types of nodes could
produce foci in perturbations and the first type of nodes is also
involved in the existence of invariant straight lines of
differential systems. For example it can  easily be shown that if
a quadratic differential system has two finite star nodes then
necessarily the system possesses invariant straight lines of total
multiplicity 6.

Furthermore, a generic node at infinity may or may not have the two exceptional
curves lying on the line at infinity. This leads to two different
situations for the phase portraits. For this reason we split the
generic nodes at infinity in two types.

The distinctions among the nilpotent and linearly zero singularities
finite or infinite can also be refined, as done in \cite[Section
4]{Art-Llib-Schl-Vlp-RMJM-CRM Report}.

The \textit{geometric equivalence relation} for finite or infinite
singularities,  introduced in \cite{Art-Llib-Schl-Vlp-RMJM-2013}
and used in \cite{Art-Llib-Schl-Vlp-RMJM-CRM Report},
takes into account such distinctions. The concept of
 \textit{geometric equivalence} of configurations of singularities
was defined and discussed in detail in a full section (Section 4)
of our paper \cite{Art-Llib-Schl-Vlp-RMJM-CRM Report},
also in \cite{Art-Llib-Schl-Vlp-RMJM-2013}.
This concept involves several notions such as ``tangent equivalence'',
``order equivalence of weak singularities'' and
``blow-up equivalence''. This last notion is subtle and cannot be described briefly.
Therefore we advise the interested reader to consult
Section 4 of \cite{Art-Llib-Schl-Vlp-RMJM-CRM Report} or of
\cite{Art-Llib-Schl-Vlp-RMJM-2013}.

This equivalence relation is deeper than the \textit{qualitative
equivalence relation} introduced by Jiang and Llibre in \cite{J_L}
because it distinguishes among the foci (or saddles) of different
orders and among the various types of nodes. This equivalence
relation also induces a deeper distinction among the more
complicated degenerate singularities.

To distinguish among the foci (or saddles) of various orders we use
the algebraic concept of Poincar\'e-Lyapunov constants. We call
strong focus (or strong saddle) a focus (or a saddle) with non--zero
trace of the linearization matrix at this point. Such a focus (or
saddle) will be considered to have the order zero. A focus (or
saddle) with trace zero is called a weak focus (weak saddle). For
details on Poincar\'e-Lyapunov constants and weak foci we refer to
\cite{Schlomiuk:1993b}, \cite{L_S}.

Algebraic information may not be significant for the local (topological)
phase portrait around a singularity. For example, topologically there is
no distinction between a focus and a node or between a weak and a
strong focus. However, as indicated before, algebraic information
plays a fundamental role in the study of perturbations of systems
possessing such singularities.


The following is a legitimate question:
\begin{quote}
How far can we go in the global theory of quadratic (or more generally
polynomial) vector fields by using
mainly algebraic means?
\end{quote}

For certain subclasses of quadratic vector fields the full
description of the phase portraits as well as of the bifurcation
diagrams can be obtained using only algebraic tools. Examples of
such classes are:

\begin{itemize}
\item the  quadratic vector fields possessing a
 center  \cite{Vulpe-DU,Schlomiuk:1993,Zoladek:1994,P-S};
\item the   quadratic Hamiltonian vector fields
 \cite{Art-Llib,ArtLliVul-EJDE};
\item the  quadratic vector fields with invariant
straight lines of total multiplicity at least four
\cite{SchVul08-RMJM,SchVul08-BASM};
\item the   planar quadratic differential systems
possessing a line of singularities at infinity \cite{Dana-Vlp-JDDE};
\item the   quadratic vector fields possessing an integrable saddle
\cite{ArtLliVul-Prep2011}.
\item the family of Lotka-Volterra systems
 \cite{Dana-Vlp-JFPT-2010,Dana-Vlp-EJDE-2012},
once we assume Bautin's analytic result saying that such systems
have no limit cycles;
\end{itemize}

In the case of other subclasses of the quadratic class
\text{QS}, such as the subclass of systems with a weak focus of
order 3 or 2 (see \cite{L_S,Art-Llib-Schl-IJBCh}) the bifurcation
diagrams were obtained by using an interplay of algebraic,
analytic and numerical methods. These subclasses were of
dimensions 2 and 3 modulo the action of the affine group and time
rescaling.  So far no 4-dimensional subclasses of \text{QS} were
studied globally so as to also produce bifurcation diagrams and
such problems are very difficult due to the number of parameters
as well as the increased complexities of these classes.

Although we now know that in trying to understand these systems,
there is a limit to the power of algebraic methods, these methods
have not been used far enough. For example the global
classification  of singularities, finite and infinite, using the
\textit{geometric equivalence relation}, \emph{can} be done by
using only algebraic methods.  The first step in this direction
was done in \cite{Art-Llib-Schl-Vlp-RMJM-2013} where the study of
\textit{the whole class}  \text{QS}, according to the
configurations of the singularities at infinity  was obtained by
using only algebraic methods. This classification was done with
respect to the \textit{geometric equivalence relation} of
configurations of singularities. Our work in
\cite{Art-Llib-Schl-Vlp-RMJM-2013} can be extended so as to also
include the finite singularities   for the whole class
\text{QS}.   To obtain the global \textit{geometric
classification} of all possible configurations of singularities,
finite and infinite, of the class \text{QS}, by purely algebraic
means is a long term goal since we expect to finally obtain over
1000 distinct configurations of singularities. In
\cite{Art-Llib-Schl-Vlp-RMJM-CRM Report} we initiated the work on
this project by studying the configurations of singularities for
the subclass of \text{QS} for which the total multiplicity $m_f$
of finite singularities is less than or equal to one.

Our goal here is to continue this work by \textit{geometrically}
classifying the configurations of all singularities with total
finite multiplicity $m_f=2$ for systems in \text{QS}.


We recall here below the notion of \textit{geometric configuration of
singularities}  defined in \cite{Art-Llib-Schl-Vlp-RMJM-CRM Report}
for both finite and infinite singularities.
We distinguish two cases:

(1) If we have a finite number of infinite singular points and a
finite  number of finite singularities we call
\textit{geometric configuration of singularities}, finite and infinite,  the
set of all these singularities each endowed with its own
multiplicity together with their local phase portraits endowed
with additional geometric structure involving the concepts of
tangent, order and blow--up equivalences defined in Section 4
of \cite{Art-Llib-Schl-Vlp-RMJM-CRM Report}
and using the notations described here in Section
\ref{sec:notation}.

(2) If the line at infinity $Z=0$ is filled up with singularities,
in each one of the charts at infinity $X\ne0$ and $Y\ne0$, the
corresponding  system in the Poincar\'e compactification (see
Section 2) is degenerate and we need to do a rescaling of an
appropriate degree of the system, so that the degeneracy be
removed. The resulting systems have only a finite number of
singularities on the line $Z=0$. In this case we call
\textit{geometric configuration of singularities}, finite and
infinite, the union of the set of all points at infinity (they are
all singularities) with the set of finite singularities - taking
care to single out the singularities at infinity of the
``reduced'' system, taken together with the local phase portraits
of finite singularities endowed with additional geometric
structure as above and the local phase portraits of the infinite
singularities of the reduced system.

We define the following affine invariants: Let $\Sigma_C$ be the
sum  of the finite orders of weak singularities (foci or weak
saddles) in a configuration $C$ of a quadratic system and let
$M_C$ be the maximum finite order of a weak singularity in a
configuration $C$ of a quadratic system. Clearly $\Sigma_C$ and
$M_C$ are affine invariants. Let $\Sigma_2$ (respectively $M_2$)
be the maximum of all $\Sigma_C$ (respectively $M_C$) for the
subclass of \text{QS} with $m_f=2$.

In stating our theorem we take care to include the results about
the configurations containing centers and integrable saddles or
containing weak singularities which are foci or saddles, since
these singularities  are especially important having the potential
of producing limit cycles in perturbations. We use the notation
introduced in \cite{Art-Llib-Schl-Vlp-RMJM-CRM Report} denoting by
$f^{(i)}$, $s^{(i)}$, the weak foci and the weak saddles of order
$i$ and by $c$ and ${\scriptstyle\$}$ the centers and integrable saddles.

Our results are stated in the following theorem.

\begin{theorem} \label{MainTheorem}
 (A) We consider
here all configurations of singularities, finite and infinite, of
quadratic vector fields with finite singularities of total
multiplicity $m_f=2$.  These configurations are classified in
Diagrams \ref{diagr:GCS-mf2-Uneg}--\ref{diagr:GCS-mf2-Uzero} according to
the geometric equivalence relation. We have 197 geometric distinct
configurations of singularities, finite and infinite. More
precisely 16 configurations with two distinct complex  finite
singularities; 151 configurations with two distinct real finite
singularities  and 30 with one real finite singularity of
multiplicity $2$.
\smallskip

(B) For the subclass of \text{QS} with $m_f=2$ we have $\Sigma_2=2=M_2$.
There are only 6 configurations of singularities with  finite weak
singular points with $\Sigma_C=2$. These have the following combinations
of finite singularities: $f^{(1)},f^{(1)}$;  $s^{(1)},s^{(1)}$;
 $s^{(2)},n$; $s^{(2)},n^d$; $s^{(2)},f$; $f^{(2)},s$.

There are 7 configurations of singularities with  finite weak singular points
with $\Sigma_C=1$. These have the following combinations of finite singularities:
$f^{(1)},n$; $f^{(1)},n^d$; $f^{(1)},s$; $f^{(1)},f$; $s^{(1)},n$;
 $s^{(1)},n^d$; $s^{(1)},f$.

There are 19 configurations containing a center or an integrable
saddle, only 6 of them with a center. There are 8 distinct couples
of finite singularities occurring in these configurations.  They
are: $c$,${\scriptstyle\$}$; $c,s$; ${\scriptstyle\$}$, ${\scriptstyle\$}$; ${\scriptstyle\$}$,$s$; ${\scriptstyle\$}$,$n$;
${\scriptstyle\$}$,$n^*$; ${\scriptstyle\$}$,$n^d$; ${\scriptstyle\$}$,$f$.
\smallskip

(C) Necessary and sufficient conditions for each one of
the 197 different equivalence classes can be assembled from these
diagrams in terms of 31 invariant polynomials with respect to
the action of the affine group and time rescaling, given in
Section~\ref{sec:Invariant polyn.}.
\smallskip

(D) The {Diagrams
\ref{diagr:GCS-mf2-Uneg}--\ref{diagr:GCS-mf2-Uzero}} actually
contain the global bifurcation diagram in the 12-dimensional space
of parameters, of the global configurations of singularities,
finite and infinite, of this family of quadratic differential
systems.
\smallskip

(E) Of all the phase portraits in the neighborhood  of
the line at infinity, which are here given in {Figure
\ref{fig:top_dis_inf}}, six are not realized in the family of
systems with $m_f=2$. They are \textit{Configs}  17; 19; 30; 32;
43; 44. (see {Figure \ref{fig:top_dis_inf}}).
\end{theorem}

\begin{remark} \label{rmk1} \rm
The diagrams are constructed using the invariant  polynomials $\mu_0$,
$\mu_1$,\dots
which are defined in Section 5. In the diagrams conditions on these
invariant polynomials are listed on the left side of the diagrams,
 while the specific geometric configurations appear on the right side
of the diagram. These configurations are expressed using the notation
described in Section 4.
\end{remark}


\begin{table}[ht]
\begin{center}
\includegraphics[width=0.98\textwidth]{diagram1} % EPS-Files/Global-mf2-Uneg-2.eps
\end{center}
\caption{Global configurations: the case $\mu_0=\mu_1=0$,
  $\mu_2\ne0 $, $\mathbf{U}<0$.}\label{diagr:GCS-mf2-Uneg}
\end{table}

\begin{table}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth, height=0.8\textheight]{diagram2a}
%Global-mf2-Upoz-a2.eps
\end{center}
\caption{Global configurations: the case $\mu_0=\mu_1=0$,
  $\mu_2\ne0$, $\mathbf{U}>0$.}\label{diagr:GCS-mf2-Upoz}
\end{table}


\begin{table}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth, height=0.8\textheight]{diagram2b}
 %Global-mf2-Upoz-b2.eps
\\
\textsc{Diagram} \ref{diagr:GCS-mf2-Upoz} ({\it continued}).
Global configurations: the case $\mu_0=\mu_1=0$,
  $\mu_2\ne 0$, $\mathbf{U}>0$.
\end{center}
\end{table}

\begin{table}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth, height=0.8\textheight]{diagram2c}
 % Global-mf2-Upoz-c4.eps
\\
\textsc{Diagram} \ref{diagr:GCS-mf2-Upoz} ({\it continued}).
Global configurations: the case $\mu_0=\mu_1=0$,
  $\mu_2\ne 0$, $\mathbf{U}>0$.
\end{center}
\end{table}

 \begin{table}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth, height=0.8\textheight]{diagram2d}
 % Global-mf2-Upoz-d4.eps
\\
\textsc{Diagram} \ref{diagr:GCS-mf2-Upoz} ({\it continued}).
Global configurations: the case $\mu_0=\mu_1=0$,
  $\mu_2\ne 0$, $\mathbf{U}>0$.
\end{center}
\end{table}

\begin{table}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth, height=0.8\textheight]{diagram2e}
 % Global-mf2-Upoz-e4.eps
\\
 \textsc{Diagram} \ref{diagr:GCS-mf2-Upoz} ({\it continued}).
Global configurations: the case $\mu_0=\mu_1=0$,
  $\mu_2\ne 0$, $\mathbf{U}>0$.
\end{center}
\end{table}



\begin{table}[ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{diagram2f}
% Global-mf2-Upoz-f3.eps
\\
\textsc{Diagram} \ref{diagr:GCS-mf2-Upoz} ({\it continued}).
Global configurations: the case $\mu_0=\mu_1=0$,
  $\mu_2\ne 0$, $\mathbf{U}>0$.
\end{center}
\end{table}


\begin{table}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth, height=0.8\textheight]{diagram3}
% Global-mf2-Uzero-a2.eps
\end{center}
\caption{Global configurations: the case $\mu_0=\mu_1=0$,
  $\mu_2\ne 0$, $\mathbf{U}=0$.}
\label{diagr:GCS-mf2-Uzero}
\end{table}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth, height=0.8\textheight]{fig2}
% Figures_1-46-c.eps
\end{center}
 \caption{Topologically distinct local configurations of \text{ISP}s
 (\cite{Dana-Vlp-JDE},\cite{Dana-Vlp-JDDE}).} \label{fig:top_dis_inf}
\end{figure}


The invariants and comitants of differential equations used   for
proving our main results are obtained following the theory of
algebraic invariants of polynomial differential systems, developed
by Sibirsky and his disciples (see for instance
\cite{Sib1,Vlp1,Popa,Baltag,Calin}).

\begin{remark} \label{rmk2} \rm
 We note that the geometric equivalence relation for
configurations is much deeper than the topological equivalence.
Indeed, for example the topological equivalence does not
distinguish between the following three configurations which are
geometrically non-equivalent: $n$, $f$, $SN$, $\copyright$,
$\copyright$; $n$, $f^{(1)}$, $SN$,
$\copyright$, $\copyright$ and
$n^d,\, f^{(1)},\, SN$, $\copyright$, $\copyright$
 where $n$ means
a singularity which is a node, capital letters indicate points at
infinity, $\copyright$ in case of a complex point and $SN$ a
saddle--node at infinity.
\end{remark}


\section{Compactifications associated to planar polynomial
differential systems}\label{sec:compact}

\subsection{Compactification on the sphere and on the Poincar\'e
disk}  Planar polynomial differential systems \eqref{il1} can be
compactified on the sphere.  For this we consider the affine plane
of coordinates $(x,y)$ as being the plane $Z=1$ in $\mathbb{R}^3$ with the
origin located at $(0,0,1)$, the $x$--axis parallel with the
$X$--axis in $\mathbb{R}^3$, and the $y$--axis parallel to the $Y$--axis.
We use a central projection to project this plane on the sphere as
follows: for each point $(x,y,1)$ we consider the line joining the
origin with $(x,y,1)$. This line intersects the sphere in two
points $P_1=(X,Y,Z)$ and $P_2=(-X,-Y,-Z)$ where
$(X,Y,Z)=(1/\sqrt{x^2+y^2+1})(x,y,1)$. The applications
$(x,y)\mapsto P_1$ and $(x,y)\mapsto P_2$ are bianalytic and
associate to a vector field on the plane $(x,y)$ an analytic
vector field $\Psi$ on the upper hemisphere and also an analytic
vector field $\Psi'$on the lower hemisphere. A theorem stated by
Poincar\'e and proved in \cite{G-V} says that there exists an
analytic vector field $\Theta$ on the whole sphere which
simultaneously extends the vector fields on the two hemispheres.
By the \textit{Poincar\'e compactification on the sphere} of a
planar polynomial vector field we mean the restriction
$\bar{\Psi}$ of the vector field $\Theta$ to the union of the
upper hemisphere with the equator.  For more details we refer to
\cite{Dum-Llib-Art}. The vertical projection of $\bar{\Psi}$ on
the plane $Z=0$ gives rise to an analytic vector field $\Phi$ on
the unit disk of this plane. By the \textit{compactification on
the Poincar\'e disk} of a planar polynomial vector field we
understand the vector field $\Phi$. By a \textit{singular point at
infinity} of a planar polynomial vector field we mean a singular
point of the vector field $\bar{\Psi}$ which is located on the
equator of the sphere, respectively a singular point of the vector
field $\Phi$ located on the boundary circle of the Poincar\'e
disk.

\subsection{Compactification on the projective plane}

To  polynomial system \eqref{il1} we can associate a differential
equation $\omega_1=q(x,y)dx-p(x,y)dy=0$. Since the differential
system \eqref{il1} is with real coefficients, we may associate to
it a foliation with singularities on the real, respectively
complex, projective plane as indicated below. The equation
$\omega_1=0$ defines a foliation with singularities on the real or
complex plane depending if we consider the equation as being
defined over the real or complex affine plane. It is known that we
can compactify these foliations with singularities on the real
respectively complex projective plane. In the study of real planar
polynomial vector fields, their associated complex vector fields
and their singularities play an important role. In particular such
a vector field could have complex, non-real singularities, by this
meaning singularities of the associated complex vector field. We
briefly recall below  how these foliations with singularities are
defined.

The application $\Upsilon: \mathbb{K} ^2\longrightarrow P_2(\mathbb{K})$ defined by
$(x,y)\mapsto [x:y:1]$ is an injection of the plane $\mathbb{K}^2$ over the
field $\mathbb{K}$ into the projective plane $P_2(\mathbb{K})$ whose image is the
set of $[X:Y:Z]$ with $Z\neq 0$. If $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$ this
application is an analytic injection. If $Z\neq 0$ then
$(\Upsilon)^{-1}([X:Y:Z])=(x,y)$ where $(x,y)=(X/Z,Y/Z)$. We obtain
a map $i:\mathbb{K}^3\setminus\{Z=0\}\longrightarrow \mathbb{K}^2$ defined by
$[X:Y:Z]\mapsto (X/Z,Y/Z)$.

Considering that $dx=d(X/Z)=(ZdX-XdZ)/Z^2$ and $dy=(ZdY-YdZ)/Z^2$,
the pull-back of the form $\omega_1$ via the map $i$ yields the form
\[
i_*(\omega_1)=q(X/Z,Y/Z)(ZdX-XdZ)/Z^2-p(X/Z,Y/Z)(ZdY-YdZ)/Z^2
\]
which has poles on $Z=0$. Then the form
$\omega=Z^{m+2}i_*(\omega_1)$ on $K^3\setminus\{Z=0\}$, $K$ being
$\mathbb{R}$ or $\mathbb{C}$ and $m$ being the degree of
 systems \eqref{il1} yields the equation $\omega=0$:
\begin{equation}
A(X,Y,Z)dX+B(X,Y,Z)dY+C(X,Y,Z)dZ=0
\end{equation}
on $K^3\setminus\{Z=0\}$
where $A$, $B$, $C$ are homogeneous polynomials over $K$ with
\begin{gather*}
A(X,Y,Z)=ZQ(X,Y,Z), \quad Q(X,Y,Z)=Z^mq(X/Z,Y/Z), \\
B(X,Y,Z)=ZP(X,Y,Z), \quad P(X,Y,Z)=Z^mp(X/Z,Y/Z),\\
C(X,Y,Z)=YP(X,Y,Z)-XQ(X,Y,Z).
\end{gather*}
The equation $AdX+BdY+CdZ=0$ defines a foliation \textit{F} with
singularities on the projective plane over $K$ with $K$ either $\mathbb{R}$
or $\mathbb{C}$. The \textit{points at infinity} of the foliation defined by
$\omega_1=0$ on the affine plane are the points $[X:Y:0]$ and the
line $Z=0$ is called the \textit{line at infinity} of the foliation
with singularities generated by $\omega_1=0$.

The singular points of the foliation \textit{F} are the solutions of
the three equations $A=0$, $B=0$, $C=0$.  In view of the definitions
of $A,B,C$ it is clear that the singular points at infinity are the
points of intersection of $Z=0$ with $C=0$.

\subsection{Assembling data on infinite singularities in divisors
of the line at infinity}

 In the previous sections we have seen
that there are two types of multiplicities for a singular point
$p$ at infinity: one expresses the maximum number $m$ of infinite
singularities which can split from $p$, in small perturbations of
the system and the other expresses the maximum number $m'$ of
finite singularities which can split from $p$, in small
perturbations of the system. We shall use a column $(m',m)^t$ to
indicate this situation.

We are interested in the global picture which includes \textit{all}
singularities at infinity.  Therefore we need to assemble the data
for individual singularities in a convenient, precise way. To do
this we use for this situation the notion of \textit{cycle} on an
algebraic variety as indicated in \cite{P-S} and which was used in
\cite{L_S} as well as in \cite{Dana-Vlp-JDE}.

We briefly recall here the definition of cycle. Let $V$ be an
irreducible algebraic variety over a field $\mathbb{K}$. A \textit{cycle} of
dimension $r$ or $r-cycle$ on $V$ is a formal sum $\sum_{W}n_WW$,
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 the coefficients $n_W$ are non-zero. The {\it
degree} deg($J$) of a cycle $J$ is defined by $\sum_{W}n_W$. An
$(n-1)$-cycle is called a {\it divisor} on $V$. These notions were
used for classification purposes of planar quadratic differential
systems in \cite{P-S,L_S,Dana-Vlp-JDE}.

To  system \eqref{il1} we can associate two divisors on the line at
infinity $Z=0$ of the complex projective plane:
$D_S(P,Q;Z)=\sum_wI_w(P,Q)w$ and $D_S(C,Z)=\sum_wI_w(C,Z)w$ where
$w\in \{Z=0\}$ and where by $I_w(F,G)$ we mean the intersection
multiplicity at $w$ of the curves $F(X,Y,Z)=0$ and $G(X,Y,Z)=0$,
with $F$ and $G$ homogeneous polynomials in $X,Y,Z$ over $\mathbb{C}$. For
more details see \cite{L_S}.

Following \cite{Dana-Vlp-JDE} we assemble the above two divisors on
the line at infinity into just one but  with values in the ring
$\mathbb{Z}^2$:
$$
        D_S = \sum_{\omega\in \{Z=0\}}\begin{pmatrix} I_w(P,Q)\\
        I_w(C,Z) \end{pmatrix} w.
$$
This divisor encodes the total number of singularities at infinity
of a system \eqref{il1} as well as the two kinds of multiplicities
which each singularity has. The meaning of these two kinds of
multiplicities are described in the definition of the two divisors
$D_S(P,Q;Z)$ and $D_S(C,Z)$ on the line at infinity.

\section{Some geometric concepts}\label{sec:definitions}

Firstly we recall some terminology.

\begin{itemize}
\item[] We call {\it elemental} a singular point with its both eigenvalues
not zero.

\item[] We call {\it semi--elemental} a singular point with exactly one of its eigenvalues
equal to zero.

\item[] We call {\it nilpotent} a singular point with both its eigenvalues
zero but with its Jacobian matrix at this point not identically
zero.

\item[] We call {\it intricate} a singular point with its
Jacobian matrix identically zero.
\end{itemize}

The {\it intricate} singularities are usually called in the
literature {\it linearly zero}. We use here the term {\it intricate}
to indicate the rather complicated behavior of phase curves around
such a singularity.


In this section we use the same concepts we considered in
\cite{Art-Llib-Schl-Vlp-RMJM-2013} and \cite{Art-Llib-Schl-Vlp-RMJM-CRM Report}
such as \textit{orbit $\gamma$ tangent to a semi--line $L$ at $p$},
\textit{well defined angle at $p$},
\textit{characteristic orbit at a singular point $p$},
\textit{characteristic angle at a singular point},
\textit{characteristic direction at} $p$.
Since these are basic concepts for the notion of
\textit{geometric equivalence relation} we recall here these
notions as well as a few others.

We assume that we have an isolated singularity $p$. Suppose that in
a neighborhood $U$ of $p$ there is no other singularity. Consider an
orbit $\gamma$ in $U$ defined by a solution $\Gamma(t)=(x(t),y(t))$
such that $\lim_{t\to+\infty}\Gamma(t)=p$ (or
$\lim_{t\to-\infty}\Gamma(t)=p$). For a fixed $t$ consider the unit
vector
$C(t)=(\overrightarrow{\Gamma(t)-p})/\|\overrightarrow{\Gamma(t)-p}\|$.
Let $L$ be a semi--line ending at $p$.  We shall say that
\textit{the orbit $\gamma$ is tangent to a semi--line $L$ at $p$} if
$\lim_{t\to+\infty} C(t)$ (or $\lim_{t\to-\infty} C(t)$) exists and
$L$ contains this limit point on the unit circle centered at $p$. In
this case we call \textit{well defined angle of} $\Gamma$ \textit{at
$p$} the angle between the positive $x$--axis and the semi--line $L$
measured in the counterclockwise sense.  We may also say that {\it
the solution curve $\Gamma(t)$ tends to $p$ with a well defined
angle}. A \textit{characteristic orbit at a singular point $p$} is
the orbit of a solution curve $\Gamma(t)$ which tends to $p$ with a
well defined angle. We call \textit{characteristic angle at the
singular point $p$} a well defined angle of a solution curve
$\Gamma(t)$. The line through $p$ extending the semi-line $L$ is
called a \textit{characteristic direction}.

Assume the singularity is placed at $(0,0)$.
Then the polynomial $PCD(x,y)=yp_m(x,y)-xq_m(x,y)$,
where $m$ is the starting degree of a polynomial differential system of the form
\eqref{il1}, is called the {\it Polynomial of Characteristic Directions} of
\eqref{il1}. In fact in case $PCD(x,y)\not\equiv 0$ the factorization of
$PCD(x,y)$ gives the characteristic directions at the origin.

If a singular point has an infinite number of characteristic
directions, we will call it a {\it star--like} point.

It is known that the neighborhood of any isolated singular point
of a polynomial vector field, which is not a focus or a center, is
formed by a finite number of sectors which could only be of three
types: parabolic, hyperbolic and elliptic (see
\cite{Dum-Llib-Art}). It is also known that any degenerate
singular point (nilpotent or intricate) can be desingularized by
means of a finite number of changes of variables, called
blow--up's, into elementary singular points (for more details see
the Section on blow--up in \cite{Art-Llib-Schl-Vlp-RMJM-2013} or
\cite{Dum-Llib-Art}).


Consider the three singular points given in Figure \ref{ES}.
All three are topologically equivalent and their neighborhoods can
be described as having two elliptic sectors and two parabolic ones.
But we can easily detect some geometric features which distinguish
them. For example $(a)$ and $(b)$ have three characteristic
directions and $(c)$ has only two. Moreover in $(a)$ the solution
curves of the parabolic sectors are tangent to only one
characteristic direction and in $(b)$ they are tangent to two
characteristic directions. All these properties can be determined
algebraically.


\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{fig3} % blow10a.eps
\end{center}
\caption{Some topologically equivalent singular points.}\label{ES}
\end{figure}


The usual definition of a sector is of topological nature and it
is local with respect to a neighborhood around the singular point.
We work with a new notion, namely of \textit{geometric local
sector}, introduced in \cite{Art-Llib-Schl-Vlp-RMJM-2013} which
distinguishes the phase portraits  of Figure \ref{ES}. As we
shall later see this notion is characterized in algebraic terms.

We begin with the elemental singular points having characteristic
directions. These are either two-directions nodes, one-direction
nodes, star nodes or saddles. The first  three cases are
distinguished algebraically using their eigenvalues (see Figure
\ref{Nodes}). In the case of saddles the notion of geometric local
sector coincides with usual notion of topological sector.

We consider now the semi--elemental singular points. These could
be saddles, nodes or saddle--nodes. Each saddle has four
separatrices and four hyperbolic sectors. Here again we call
geometric local sector any one of these hyperbolic sectors and we
call \textit{borsec} (contraction of border with sector) any one
of the four separatrices.

A semi--elemental node has two characteristic directions
generating four half lines. For each one of these  half lines
there exists at least one orbit tangent to that half line and we
pick an orbit tangent to that half line. Removing these four
orbits together with the singular point, we are left with four
sectors which we call \textit{geometric local sectors}  and we
call \textit{borsecs} these four orbits.

Consider now a semi--elemental saddle--node. Such a singular point
has three separatrices and three topological sectors, two
hyperbolic ones and one parabolic sector. Such a singular point
has four characteristic half lines and one of them separates the
parabolic sector in two. By removing an orbit tangent to a half
line for each one of the  half lines as well as the singular point we obtain
four sectors which we call geometric local sectors. We call
borsecs these four orbits.

We now proceed to extend the notion of  geometric local sector and
of borsec for nilpotent and intricate singular points.


The introduction of the concept of borsec in the general case will
play a role in distinguishing a semi--elemental saddle--node from
an intricate saddle--node such as the one indicate  in
Figure \ref{intricate saddlenode}. In the semi--elemental saddle--node
all orbits inside the parabolic sector are tangent to the same
half--line but in the saddle-node of Figure \ref{intricate
saddlenode} the orbits in the parabolic sector are not all tangent
to the same half--line. The orbits in this parabolic sector are of
three kinds: the ones tangent to separatrix $(a)$, the ones
tangent to separatrix $(c)$ and a single orbit which is tangent to
other half--line of the characteristic direction defined by
separatrix $(b)$. In this case this last orbit is called the
borsec. The other three borsecs are separatrices as in the case of
the semi--elemental saddle--node.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.4\textwidth]{fig4}
% blow8a.eps
\end{center}
 \caption{Local phase portrait of a non semi--elemental
saddle--node.}\label{intricate saddlenode}
\end{figure}


To extend the notion of  geometric local sector and of borsec for
nilpotent and intricate singular points we start by introducing
some terminology.



 Let $\delta$ be the border of a sufficiently small open disc $D$
centered at point $p$ so that $\delta$ intersects all the
elliptic, parabolic and hyperbolic sectors of a nilpotent or
intricate singular point $p$.

Consider a solution $\Gamma: (a,b)\rightarrow\,\mathbb{R}^2$ where
$(a,b)$ is its maximal interval of definition and let $\gamma$ be
the orbit of $\Gamma$, i.e. $\gamma=\left\{\Gamma(t)\,|\,t\in
(a,b)\right\}$. We call \textit{half orbit} of $\gamma$ at a
singular point $p$ a subset $\gamma'\subseteq \gamma$ such that
there exists $t_1\in (a,b)$ for which we have either
$\gamma'=\left\{\Gamma(t)\,|\,t\in (a,t_1)\right\}$ in which case
we  have $a=-\infty$, ${
\lim_{t\to-\infty}\Gamma(t)=p}$, $\Gamma(t_1)\in \delta$ and
$\Gamma(t)\in D$ for $t\in (-\infty,t_1)$, or
$\gamma'=\left\{\Gamma(t)\,|\,t\in (t_1,b)\right\}$, $b= +\infty$,
${ \lim_{t\to+\infty}\Gamma(t)=p}$, $\Gamma(t_1)\in
\delta$ and $\Gamma(t)\in D$ for $t\in (t_1,\infty)$.

We note that in the case of elliptic sectors there may exist
orbits which are divided exactly in two half orbits.

Let $\Omega_p=\{\gamma':\gamma' \text{ is a half orbit at } p\}$.

We shall define a relation of equivalence on $\Omega_p$ by using
the complete desingularization of the singular point $p$ in case
this point is nilpotent or intricate. There are two ways to
desingularize such a singular point: by passing to polar
coordinates or by using rational changes of coordinates. The first
has the inconvenience of using   trigonometrical functions, and
this becomes a serious problem when a chain of blow--ups are
needed in order to complete the desingularization of the
degenerate point. The second uses rational changes of coordinates,
convenient  for our polynomial systems. In such a case two
blow--ups in different directions are needed and information from
both must be glued together  to obtain the desired portrait.

Here for desingularization we use the second possibility, namely
with rational changes of coordinates  at each stage of the
process. Two rational changes are needed, one for each direction
of the blow--up. If at a stage the coordinates are $(x,y)$ and we
do a blow--up of a singular point in $y$-direction, this means
that we introduce a new variable $z$ and consider the diffeomorphism
of the $(x,y)$ plane for $x\neq 0$ defined by $\phi(x,y)=(x,y,z)$ where $y=xz$.
This diffeomorphism
transfers our vector field on the subset $x\neq 0$ of the plane $(x,y)$
on the subset $x\neq 0$ of the algebraic surface $y=zx$.
It can easily be checked that the projection $(x,xz,z)\mapsto (x,z)$ of
this surface on the $(x,z)$ plane is a diffeomorphism. So our vector
field on the plane $(x,y)$ for $x\neq 0$ is diffeomeorphic to the vector
field thus obtained on the $(x,z)$ plane for $x\neq 0$. The singular
point $(x_0,y_0)$
which we can assume to be placed at the origin $(0,0)$, is then
replaced by the straight line $x=0=y$ in the 3-dimensional space of
coordinates $x,y,z$. This line is also the $z$-axis of the plane
$(x,z)$ and it is called \textit{blow--up line}.

Analogously we can do a blow-up in the $x$-direction  using the
change $(x,y)\to(zy,y)$ which  is a diffeomorphism for $y\ne0$.


The two directional blow--ups can be simplified in just one
1--direction blow--up if we make sure   that the direction in
which we do a blow--up is not a characteristic direction, so as to
be sure  that we are not going to lose information doing the
blow--up in the chosen direction. This can be easily solved by a
simple linear change of coordinates of the type $(x,y)\to(x+k
y,y)$ where $k$ is a constant (usually 1). It seems natural to
call this linear change a {\it k--twist} as the $y$--axis gets
twisted with some angle depending on $k$. It is obvious that the
phase portrait of the degenerate point which is studied  cannot
depend on the set of $k$'s used in the desingularization process.

 Since the complete desingularization of a nilpotent or an
intricate singular point in general needs more than one blow--up,
we have as many blow--up lines as we have blow--ups. As indicated
above a blow--up line may be transformed by means of linear
changes and through other   blow--up's in other straight lines. We
will call such straight lines {\it blow--up lines of higher order}.

We now introduce an  equivalent relation  on $\Omega_p$.  We say
that two half orbits $\gamma'_1,\,\gamma'_2\in\Omega_p$ are
equivalent if and only if (i) for both $\gamma'_1$ and
$\gamma'_2$ we have  ${
\lim_{t\to-\infty}\Gamma_1(t)=p=\lim_{t\to-\infty}\Gamma_2(t)}$ or
${\lim_{t\to+\infty}\Gamma_1(t)=p=\lim_{t\to+\infty}\Gamma_2(t)}$,
and (ii) after the complete desingularization, these orbits
lifted to the final stage are tangent to the same half--line at
the same singular point, or end as orbits of a star node on the
same half--plane defined by the blown--up line, and (iii)  both
orbits must remain in the same half--plane in all the successive
blow--up's.

We recall that after a complete desingularization all singular
points are elemental or semi--elemental. We now single out two
types of equivalence classes:

(a) Suppose that an equivalence class $\mathcal C\in
\Omega_p/\sim$ is such that its half orbits lifted to the last
stage in the desingularization process lead to orbits which
possess the following properties:\ $i)$ they belong to an
elemental two--directions node or to a semi--elemental
saddle--node, and \ $ii)$ they are all tangent to the same
half--line which lies on the blow--up line.

  (b) Suppose that an equivalence class $\mathcal{ C}\in
\Omega_p/\sim$ is such that  (i) its half orbits lifted to
the final stage of the desingularization process, are tangent to a
blow--up line of higher order, and  (ii) its lifted orbits
blown--down to the previous stage of the desingularization, form a
part of an elliptic sector.

Let $\Omega'_p/\sim$ be the set of all equivalence classes which
are of type $(a)$ or $(b)$. Then consider the complement
$\mathcal{B}_p= (\Omega_p/\sim)-(\Omega'_p/\sim)$ and consider
a set of representatives of $\mathcal{B}_p$. We call {\it borsec}
anyone of these representatives.

  Note that the definition of borsec is independent of the
choice  of the disc $D$ with boundary $\delta$ if $D$ is
sufficiently small.


We call {\it geometric local sector} of a singular point $p$ with
respect to a neighborhood $V$,   a region in $V$ delimited by two
consecutive borsecs.


To illustrate the definitions of borsec and geometric
local sector we will discuss the following example given in
Figures \ref{intricate}, \ref{intricateblow}A and
\ref{intricateblow}B.

\begin{figure}[ht]
% \psfrag{A}{$a$} \psfrag{B}{$b$} \psfrag{C}{$c$} \psfrag{D}{$d$}
% \psfrag{E}{$e$} \psfrag{F}{$f$} \psfrag{G}{$g$} \psfrag{H}{$h$}
% \psfrag{I}{$i$} \psfrag{J}{$j$} \psfrag{K}{$k$} \psfrag{L}{$l$}
% \psfrag{Q1}{$q_1$} \psfrag{Q2}{$q_2$} \psfrag{Q3}{$q_3$} \psfrag{Q4}{$q_4$}
%  \psfrag{Q5}{$q_5$} \psfrag{R1}{$r_1$} \psfrag{R2}{$r_2$} \psfrag{R3}{$r_3$}
%  \psfrag{R4}{$r_4$}
% \psfrag{X}{$A$} \psfrag{Y}{$B$}
% \psfrag{S01}{$\emptyset_1$} \psfrag{S02}{$\emptyset_2$}
% \psfrag{S20}{$20^*$} \psfrag{S21}{$21^*$} \psfrag{S22}{$22$} \psfrag{S23}{$23^*$} \psfrag{S24}{$24$}
% \psfrag{S10}{$10^*$} \psfrag{S1_1}{$11$} \psfrag{S12}{$12^-$} \psfrag{S13}{$13$} \psfrag{S14}{$14^*$}% \psfrag{S15}{$15$}% \psfrag{S16}{$16^-$}% \psfrag{S17}{$17^*$}% \psfrag{S18}{$18^-$}% \psfrag{S19}{$19^-$}
% \psfrag{S1}{$1^-$} \psfrag{S2}{$2$} \psfrag{S3}{$3^*$} \psfrag{S4}{$4^*$} \psfrag{S5}{$5$}% \psfrag{S6}{$6^-$}% \psfrag{S7}{$7^-$}% \psfrag{S8}{$8$}% \psfrag{S9}{$9^*$}
\begin{center}
\includegraphics[width=0.98\textwidth]{fig5}
% blow1a-new.eps
\end{center}
\caption{Local phase portrait of an intricate singular point.}\label{intricate}
\end{figure}

\begin{figure}[ht]
% \psfrag{A}{$a$} \psfrag{B}{$b$} \psfrag{C}{$c$} \psfrag{D}{$d$}
% \psfrag{E}{$e$} \psfrag{F}{$f$} \psfrag{G}{$g$} \psfrag{H}{$h$}
% \psfrag{I}{$i$} \psfrag{J}{$j$} \psfrag{K}{$k$} \psfrag{L}{$l$}
% \psfrag{Q1}{$q_1$} \psfrag{Q2}{$q_2$} \psfrag{Q3}{$q_3$} \psfrag{Q4}{$q_4$}
% \psfrag{Q5}{$q_5$} \psfrag{R1}{$r_1$} \psfrag{R2}{$r_2$} \psfrag{R3}{$r_3$} \psfrag{R4}{$r_4$}
% \psfrag{X}{$A$} \psfrag{Y}{$B$}
% \psfrag{S01}{$\emptyset_1$} \psfrag{S02}{$\emptyset_2$}
% \psfrag{S20}{$20^*$} \psfrag{S21}{$21^*$} \psfrag{S22}{$22$} \psfrag{S23}{$23^*$}
% \psfrag{S24}{$24$}
% \psfrag{S10}{$10^*$} \psfrag{S1_1}{$11$} \psfrag{S12}{$12^-$} \psfrag{S13}{$13$}
% \psfrag{S14}{$14^*$} \psfrag{S15}{$15$} \psfrag{S16}{$16^-$} \psfrag{S17}{$17^*$}
% \psfrag{S18}{$18^-$} \psfrag{S19}{$19^-$}
% \psfrag{S1}{$1^-$} \psfrag{S2}{$2$} \psfrag{S3}{$3^*$} \psfrag{S4}{$4^*$}
% \psfrag{S5}{$5$} \psfrag{S6}{$6^-$} \psfrag{S7}{$7^-$} \psfrag{S8}{$8$}
% \psfrag{S9}{$9^*$}
\begin{center}
\includegraphics[width=0.98\textwidth]{fig6} %  blow1b.eps
\\ A\hfil B
\end{center}
\caption{The two needed blow--ups for point of Figure \ref{intricate}.}
\label{intricateblow}
\end{figure}

We have portrayed an intricate singular point $p$ whose
desingularization needs a chain of two   blow--ups and where all
different kinds of elemental singular points and semi--elemental
saddle--nodes appear in every possible position with respect of
the blow--up line.

We have taken a small enough neighborhood of the point $p$ of
boundary $\delta$. We split the boundary $\delta$ in different
arcs and points which will correspond to the different equivalence
classes of orbits. We have enumerated them from 1 to 24. The arcs
of $\delta$ denoted with $\emptyset_1$ and $\emptyset_2$
correspond to hyperbolic sectors which are not considered in the
equivalence classes since the orbits do  not tend to $p$. Some of
these equivalence classes have a unique orbit which is then a
borsec (like $14^*$ or $4^*$). We  add an asterisk superscript to
denote these equivalence classes. Other equivalence classes are
arcs, like $16^-$ or $12^-$, and one representative of each one of
them is taken as a borsec. We   add  a dash superscript to denote
these equivalence classes. The remaining equivalence classes, just
denoted by their number, are those which do not produce a borsec
by the exceptions given in the definition. We have drawn the
separatrices (which are always borsecs) with a bold continuous
line. We have drawn the borsecs which are not separatrices with
bold dashed  lines. Other orbits are drawn as thin continuous
lines. Finally, the vertical dashed line is the $y$-direction in
which the first blow-up was done.


We  describe a little the blow--ups of the phase portrait of the
intricate point $p$ given in Figure \ref{intricate}. Its
first blow--up is given in Figure \ref{intricateblow}A. In
it we see from the upper part of the figure   to its lower part:
$q_1$) an elemental two--directions node with all but two orbits
tangent to the blow--up line; $q_2$) a semi--elemental
saddle--node with direction associated to the non--zero eigenvalue
being the blow--up line; $q_3$) another intricate singular point
which   needs another blow--up portrayed in Figure
\ref{intricateblow}B; $q_4$) an elemental saddle; and $q_5$) an
elemental one--direction node which necessarily has its
characteristic direction  coinciding with the blow--up line.

In order to make the vertical blow--up of the intricate point $q_3$
we must first do an $\varepsilon$--twist since the vertical
direction which corresponds to the previous blow--up line is a
characteristic direction of $q_3$.

In this second blow--up given in Figure \ref{intricateblow}B
we see going down from its upper part,   the following elemental
or semi--elemental singular points: $r_1$) a two--directions node
with only two orbits tangent to the blow--up line (this singular
point corresponds to the characteristic direction given by the
previous blow--up line); $r_2$) a saddle; $r_3$) a saddle--node
with the direction associated to the zero eigenvalue being the
blow--up line; $r_4$) a star node.

Now we describe all the classes of equivalence that we obtain in
order to clarify the definitions of borsec and geometric local
sector.

We must move from the second blow--up to the first and after that
to the original phase portrait. We enumerate the arcs in the
boundary of Figure \ref{intricateblow}B (following the
clockwise sense) which will correspond to the classes of
equivalence of orbits in Figure \ref{intricate} as follows.

($1^-$) The arc $1^-$ goes from the point $a$ on the vertical axis
to the point $b$ without including any of them.

($2$)  The arc $2$ goes from the point $b$ to the point $3^*$
without including any of them.

The orbit that ends at point $b$ corresponds to the blow--up line
in the Figure \ref{intricateblow}A, and so does not survive
in the original phase portrait. Thus the orbits associated to arc
$1^-$ cannot belong to the same equivalence class as the orbits
associated to arc 2 since in Figure \ref{intricateblow}A
they are in different half--planes defined by the blow--up line.

($3^*$) The point $3^*$ belongs to the orbit which is a separatrix
of the saddle $r_2$.

($\emptyset_1$) The open arc  $\emptyset_1$ goes between the points
$3^*$ and $4^*$ and it is associated to a hyperbolic sector and plays no
role.

($4^*$) The point $4^*$ belongs to the orbit which is a separatrix
of the saddle--node $r_3$.

($5$)  The arc $5$ goes from the point $4^*$ to the point  $c$
including only the second.

($6^-$) The arc $6^-$ goes from the point $c$ to the point $d$ on
the vertical axis, including the point $c$.

The point $c$ belongs to both arcs $5$ and $6^-$. In fact it is
just a point of partition of the boundary, splitting the orbits
that come from $r_3$ from the orbits that go to $r_4$. Since the
equivalence classes are defined regarding the half orbits there is
no contradiction.


($7^-$)  The arc $7^-$ goes from the point $d$ on the vertical axis
to the point $e$ including the point $e$ (i.e. $7^-=(d,e]$\,).

($8$)  The arc $8$ goes from the point $e$ to the point $9^*$
including the point $e$.

The same comment made for the point $c$ applies to point $e$.

($9^*$) The point $9^*$ belongs to the orbit which is a separatrix
of the saddle--node $r_3$.

($\emptyset_2$) The open arc $\emptyset_2$ between the points $9^*$
and $10^*$ is associated to a hyperbolic sector and plays no role.

($10^*$) The point $10^*$ belongs to the orbit which is a
separatrix of the saddle $r_2$.

($11$)  The arc $11$ goes from the point $10^*$ to the point $f$
without including any of them.


($12^-$)  The arc $12^-$ goes from the point $f$ to the point
$a$ in the vertical axis without including any of them (i.e.
$12^-=(d,e)$\,).


The same comment done for the point $b$ applies to point $f$.

Now we translate these notations to Figure
\ref{intricateblow}A and complete the notation of the arcs on the
boundary of this figure again following the clockwise sense.

($13$)  The arc $13$ goes from the point $g$ on the vertical axis
to the point $14^*$ without including any of them.

($14^*$) The point $14^*$ belongs to the orbit which is tangent to
the eigenvector associated to the greatest eigenvalue of the node
$q_1$.

($15$)  The arc $15$ goes from the point $14^*$ to the point $h$
including only the second.

($16^-$)  The arc $16^-$ goes from the point $h$ to the point $i$
including both (i.e. $16^-=[d,e]$\,).


The following arcs and points from the point $i$ to the point
$17^*$ have already received their names when we did the
blow--down from Figure \ref{intricateblow}B to Figure
\ref{intricateblow}A.

The arcs $6^-$ and $12^-$ of Figure \ref{intricateblow}B
become adjacent in Figure \ref{intricateblow}A and the
points $a$ and $d$ are glued together  and correspond to the point
which after the $-\varepsilon$--twist goes to the vertical axis.
The region defined by these arcs forms now an elliptic sector.

($17^*$) The point $17^*$ belongs to the orbit which is a
separatrix of the saddle $q_4$.

($18^-$)  The arc $18^-$ goes from the point $17^*$ to the point
$j$ without including any of them (i.e. $18^-=(17^*,j)$).

($19^-$)  The arc $19^-$ goes from the point $j$ to the point
$20^*$ without including any of them (i.e. $19^-=(j,20^*)$).

($20^*$) The point $20^*$ belongs to the orbit which is a
separatrix of the saddle $q_4$.

The following arcs and points from the point $20^*$ to the point
$21^*$ have already received their names when we have done the
blow--down from Figure \ref{intricateblow}B to Figure
\ref{intricateblow}A.

($21^*$) The point $21^*$ belongs to the orbit which is a
separatrix of the saddle--node $q_2$.

($22$)  The arc $22$ goes from the point $21^*$ to the point $23^*$
without including any of them.

($23^*$) The point $23^*$ belongs to the orbit which is tangent to
the eigenvector associated to the greatest eigenvalue of the node
$q_1$.

($24$) The arc $24$ goes from the point $23^*$ to the point $g$ in
the vertical axis without including any of them.

Now we move to the original phase portrait in Figure
\ref{intricate}. For clarity it is convenient to start the
description with a hyperbolic sector.


The orbit associated to the point $4^*$ defines an equivalent
class  with a single element and then, this element is a borsec.
Moreover it is a global separatrix.

The orbits associated to the points of the arc $5$ form a class of
equivalence but define no borsec since in the final
desingularization (Figure \ref{intricateblow}B) these orbits
end at a saddle--node tangent to the   blow--up line and thus
these orbits are in a class of equivalence of type $(a)$ which
does not produce borsec.

The orbits associated to the points of the arc $6^-$ form a class
of equivalence defining a borsec which splits the two local
geometric elliptic sectors that we see in Figure
\ref{intricate}. This  borsec is not a   separatrix.

The orbits associated to the points of the arc $12^-$ form a class
of equivalence defining a borsec which splits a local elliptic
sector from a parabolic local sector that we can see in Figure \ref{intricate}.
Even though the class $12^-$ has been
split from class $11$ by the blow--up line   of higher order  (the
straight line passing through point $r_1$ and going from point $b$
to point $f$  in Figure \ref{intricateblow}B), we see that
class $12^-$ corresponds to the part of an elliptic sector with
its characteristic direction tangent to the blow--up line. So,
  this class of equivalence is not of type (b)  and we must
define a borsec there. The point (b) however will occur later on
in our discussion, more precisely when we consider the arc $11$.


The orbit associated to the point $17^*$ defines an equivalent
class  with a single element and then,  this element is a borsec.
This borsec is not a separatrix. It is just part of a global
parabolic sector but locally distinguishes the three different
characteristic directions of the orbits in the arc of $\delta$
going from $d$ to $l$.

The orbits associated to the points of the arc $18^-$ form a class
of equivalence defining a borsec which splits a local elliptic
sector from a parabolic one that we can see in Figure
\ref{intricate}.

The orbits associated to the points of the arc $24$ form a class
of equivalence but this does not define a borsec because in the
final desingularization, the corresponding orbits end at a
two--directions node tangent to the blow--up line  (this class of
equivalence is of type (a)).

The orbit associated to the point $23^*$ defines an equivalent
class  with a single element and then this element is a borsec which
splits a local elliptic sector from a parabolic one that we can
see in Figure \ref{intricate}.

The orbits associated to the points of the arc $22^-$ form a class
of equivalence but this does not define a borsec because in the
final desingularization, the corresponding orbits end at a
two--directions node tangent to the blow--up line  (this class of
equivalence is of type (a)).

The orbit associated to the point $21^*$ defines an equivalent
class  with a single element and then,  this element is a borsec.

The orbits associated to the points of the arc $1^-$ form a class
of equivalence defining a borsec which splits a local elliptic
sector from a parabolic one that we can see in Figure
\ref{intricate}. Even though the  class $1^-$ has been split from
class $2$ by the blow--up line   of higher order, in Figure
\ref{intricateblow}B, we see that class $1^-$ corresponds to a
part of an elliptic sector with its characteristic direction
tangent to the blow--up line. So,   this is not a class of
equivalence of type (b) and we must define a borsec here.




The orbits associated to the points of the arc $7^-$ form a class
of equivalence defining a borsec which splits the two local
elliptic sectors that we see in Figure \ref{intricate}. As
in the case of arc $6^-$  this  borsec is not a   separatrix.

The orbits associated to the points of the arc $8$ form a class of
equivalence but define no borsec since in the final
desingularization (Figure \ref{intricateblow}B) these orbits
end at a saddle--node tangent to the blow--up line (this
equivalence class   is of type (a)).

The orbit associated to the point $9^*$ defines an equivalent
class  with a single element and then,  this element is a borsec.
Moreover it is a global separatrix.

The orbits associated to the open arc $\emptyset_2$ form a
hyperbolic sector and are not associated to any equivalence class
since  they do not end at the singular point.

The orbit associated to the point $10^*$ defines an equivalent
class  with a single element and then,  this element is a borsec.
Moreover it is a global separatrix.

  The orbits associated to the points of the arc $11$ form a
class of equivalence but define no borsec since class $11$ is of
type $(b)$. In this case we are in a similar situation as with the
arc $12^-$ but now, since the point $r_2$ is a saddle, the arc
$11$ in Figure \ref{intricateblow}A defines a parabolic
sector and so there is no need of a borsec, which would otherwise
be needed if the sector were elliptic.


The orbit associated to the point $20^*$ defines an equivalent
class  with a single element and then,  this element is a borsec.
This is similar to the case $17^*$.

The orbits associated to the points of the arc $19^-$ form a class
of equivalence defining a borsec which splits a local elliptic
sector from a parabolic one that we can see in Figure
\ref{intricate}. This is similar to the case $18^-$.

The orbits associated to the points of the arc $13$ form a class
of equivalence but this does not define a borsec analogously with
the case $24$.

The orbit associated to the point $14^*$ defines an equivalent
class  with a single element and then,  this element is a borsec.

The orbits associated to the points of the arc $15$ form a class
of equivalence which   does not define a borsec analogously to the
case $13$.

The orbits associated to the points of the arc $16^-$ form a class
of equivalence defining a borsec which splits  two local elliptic
sectors. This is similar to the case $7^-$.

The orbits associated to the points of arc $2$ form a class of
equivalence but define no borsec by the same arguments used for
the arc $11$.

The orbit associated to the point $3^*$ defines an equivalent
class  with a single element and then,  this element is a borsec.
Moreover it is a separatrix.


Generically a \textit{geometric local sector} is defined by two
borsecs arriving at the singular point with two different well
defined angles and which are consecutive. If this sector is
parabolic, then the solutions can arrive at the singular point with
one of the two characteristic angles, and this is a geometric
information than can be revealed with the blow--up.

There is also the possibility that two borsecs defining a geometric
local sector tend to the singular point with the same well defined
angle. Such a sector will be called a {\it cusp--like sector} which
can either be hyperbolic, elliptic or parabolic denoted by
$H_{\curlywedge}$, $E_{\curlywedge}$ and $P_{\curlywedge}$
respectively.

In the case of parabolic sectors we want to include the information
as the orbits arrive tangent to one or to the other borsec. We
distinguish the two cases writing by ${\overset{
{}_\curvearrowleft}P}$ if they arrive tangent to the borsec limiting
the previous sector in clockwise sense or ${\overset{
{}_\curvearrowright}P}$ if they arrive tangent to the borsec
limiting the next sector. In the case of a cusp--like parabolic
sector, all orbits must arrive with only one well determined angle,
but the distinction between ${\overset{ {}_\curvearrowleft}P}$ and
${\overset{ {}_\curvearrowright}P}$ is still valid because it
occurs at some stage of the desingularization and this can be
algebraically determined. Thus complicated intricate singular points
like the two we see in Figure \ref{vicinities} may be
described as $ \overset{{}_\curvearrowright}PE\overset{
{}_\curvearrowleft}P HHH $ (case $(a)$) and $ E {\overset{
{}_\curvearrowleft}P_{\curlywedge}}H H {\overset{
{}_\curvearrowright}P_{\curlywedge}}E $   (case $(b)$),
respectively.


\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig7}
% Vecinitiies-SP-1.eps
\end{center}
 \caption{Two phase portraits of degenerate singular points.}\label{vicinities}
\end{figure}

The phase portrait of the intricate point of Figure
\ref{intricate} could be described as
$$
H_{\curlywedge}
E_{\curlywedge} E_{\curlywedge}
\overset{{}_\curvearrowleft}P \overset{{}_\curvearrowright}P E
\overset{{}_\curvearrowleft}P \overset{{}_\curvearrowright}P
E_{\curlywedge} E_{\curlywedge} H_{\curlywedge}
\overset{{}_\curvearrowleft}P \overset{{}_\curvearrowright}P E
E E
$$
 starting with the hyperbolic sector $\emptyset_1$ and going
in the clockwise direction.


A star--like point can either be a node or something much more
complicated with elliptic and hyperbolic sectors included. In case
there are hyperbolic sectors, they must be cusp--like. Elliptic
sectors can either be cusp--like or star--like. We  call {\it
special characteristic angle} any well defined angle of a star-like
point, in which either none or more than one solution curve tends to
$p$ within this well defined angle. We will call {\it special
characteristic direction} any line such that at least one of the two
angles defining it, is a special characteristic angle.



\section{Notation for singularities of polynomial differential systems}\label{sec:notation}

 In \cite{Art-Llib-Schl-Vlp-RMJM-2013} we introduced
convenient notations which we also used in
\cite{Art-Llib-Schl-Vlp-RMJM-CRM Report} and which we are also using
here. These notations can easily be extended to general polynomial
systems.

We describe the finite and infinite singularities, denoting the
first ones with lower case letters and the second with capital
letters. When describing in a sequence both finite and infinite
singular points, we will always place first the finite ones and only
later the infinite ones, separating them by a semicolon`;'.

\textbf{Elemental points:} We use the letters `$s$',`$S$' for
``saddles''; `$n$', `$N$' for ``nodes''; `$f$' for ``foci''; `$c$'
for ``centers'' and $\scriptstyle\copyright$ (respectively
$\copyright$) for complex finite (respectively infinite)
singularities. In order to augment the level of precision we
distinguish the finite nodes as follows:

\begin{itemize}
\item{} `$n$' for a node with two distinct eigenvalues (generic node);
\item{} `$n^d$' (a one--direction node) for a node with two identical
   eigenvalues whose Jacobian matrix is not diagonal;
\item{} `$n^\ast$' (a star node) for a node with two identical
   eigenvalues whose Jacobian matrix is diagonal.
\end{itemize}

In the case of an elemental infinite generic node, we want to
distinguish whether the eigenvalue associated to the eigenvector
directed towards the affine plane is, in absolute value, greater or
lower than the eigenvalue associated to the eigenvector tangent to
the line at infinity. This is relevant because this determines if
all the orbits except one on the Poincar\'{e} disk arrive at infinity
tangent to the line at infinity or transversal to this line. We will
denote them as `$N^\infty$' and `$N^f$' respectively.

Finite elemental foci and saddles are classified as strong or weak
foci, respectively strong or weak saddles. When the trace of the
Jacobian matrix evaluated at those singular points is not zero, we
call them strong saddles and strong foci and we maintain the
standard notations `$s$' and `$f$'. But when the trace is zero,
except for centers and saddles of infinite order (i.e. with all
their Poincar\'e-Lyapounov constants equal to zero), it is known
that the foci and saddles, in the quadratic case,  may have up to 3
orders. We denote them by `$s^{(i)}$' and `$f^{(i)}$' where
$i=1,2,3$ is the order. In addition we have the centers which we
denote by `$c$' and saddles of infinite order (integrable saddles)
which we denote by `${\scriptstyle\$}$'.

Foci and centers cannot appear as singular points at infinity and
hence there is no need to introduce their order in this case. In
case of saddles, we can have weak saddles at infinity but  the
maximum order of weak singularities in cubic systems is not yet
known. For this reason, a complete study of weak saddles at
infinity cannot be done at this stage. Due to this, in
\cite{Art-Llib-Schl-Vlp-RMJM-2013} and in
\cite{Art-Llib-Schl-Vlp-RMJM-CRM Report} and here  we chose   not
even to distinguish between a saddle and a weak saddle at
infinity.


All non--elemental singular points are multiple points, in the sense
that there are perturbations which have at least two elemental
singular points as close as we wish to the multiple point. For
finite singular points we denote with a subindex their multiplicity
as in `$\overline{s}_{(5)}$' or in `$\widehat{es}_{(3)}$' (the
notation `$\overline{\phantom{s}}$' indicates that the saddle is
semi--elemental and `$\widehat{\phantom{es}}$' indicates that the
singular point is nilpotent). In order to describe the various kinds
of multiplicity for infinite singular points we use the concepts and
notations introduced in \cite{Dana-Vlp-JDE}. Thus we denote by
`$\binom{a}{b}\dots$' the maximum number $a$ (respectively $b$) of
finite (respectively infinite) singularities which can be obtained
by perturbation of the multiple point. For example
`$\overline{\binom{1}{1}}SN$' means a saddle--node at
infinity produced by the collision of one finite singularity with an
infinite one; `$\overline{\binom{0}{3}}S$' means a saddle
produced by the collision of 3 infinite singularities.

\subsection*{Semi--elemental points:} They can either be nodes, saddles or
saddle--nodes, finite or infinite. We will denote the
semi--elemental ones always with an overline, for example
`$\overline {sn}$', `$\overline {s}$' and `$\overline {n}$' with the
corresponding multiplicity. In the case of infinite points we will
put `$\overline{\phantom{s}}$' on top of the parenthesis with
multiplicities.

Moreover, in cases that will be explained later (see the paragraph
dedicated to intricate points), an infinite saddle--node may be
denoted by `$\overline{\binom{1}{1}} NS$' instead of
`$\overline{\binom{1}{1}}SN$'. Semi--elemental nodes could
never be `$n^d$' or `$n^\ast$' since their eigenvalues are always
different. In case of an infinite semi--elemental node, the type of
collision determines whether the point is denoted by `$N^f$' or by
`$N^\infty$' where `$\overline{\binom{2}{1}} N$' is an
`$N^f$' and `$\overline{\binom{0}{3}} N$' is an `$N^\infty$'.

\subsection*{Nilpotent points:} They can either be saddles, nodes,
saddle--nodes, elliptic--saddles, cusps, foci or centers. The first
four of these could be at infinity. We denote the nilpotent singular
points with a hat `$\widehat{\hphantom{n}}$' as in
$\widehat{es}_{(3)}$ for a finite nilpotent elliptic--saddle of
multiplicity 3 and $\widehat{cp}_{(2)}$ for a finite nilpotent cusp
point of multiplicity 2. In the case of nilpotent infinite points,
we will put the `$\widehat{\hphantom{n}}$' on top of the parenthesis
with multiplicity, for example $\widehat{\binom{1}{2}}\ PEP-H$
(the meaning of $PEP-H$ will be explained in next paragraph). The
relative position of the sectors of an infinite nilpotent point,
with respect to the line at infinity, can produce topologically
different phase portraits. This forces to use a notation for these
points similar to the notation which we will use for the intricate
points.

\subsection*{Intricate points:} It is known that the neighborhood of any
singular point of a polynomial vector field (except for foci and
centers) is formed by a finite number of sectors  which could only
be of three types: parabolic, hyperbolic and elliptic (see
\cite{Dum-Llib-Art}). Then, a reasonable way to describe intricate
and nilpotent points is to use a sequence formed by the types of
their sectors. The description we give is the one which appears in
the clockwise direction (starting anywhere) once the blow--down of
the desingularization is done. Thus in {\it non-degenerate}
quadratic systems (that is, both components of the system are
coprime), we have just seven possibilities for finite intricate
singular points of multiplicity four (see \cite{Art-Llib-Vlp-IJBCh})
which are the following ones: $phpphp_{(4)}$; $phph_{(4)}$;
$hh_{(4)}$; $hhhhhh_{(4)}$; $peppep_{(4)}$; $pepe_{(4)}$;
$ee_{(4)}$.

The lower case letters used here indicate that we have finite singularities
 and subindex $(4)$ indicates the multiplicity $4$ of the singularities.

For infinite intricate and nilpotent singular points, we insert a
dash (hyphen) between the sectors to split those which appear on one
side or the other of the equator of the sphere. In this way we will
distinguish between $\binom{2}{2}PHP-PHP$ and $\binom{2}{2}PPH-PPH$.


Whenever we have an infinite nilpotent or intricate singular point,
 we will always start with a sector bordering the infinity (to
avoid using two dashes). When one needs to describe a configuration
of singular points at infinity, then in some cases the relative
positions of the points, is relevant. In
\cite{Art-Llib-Schl-Vlp-RMJM-2013} this situation only occurs once
for systems having two semi--elemental saddle--nodes at infinity and
a third singular point which is elemental. In this case we need to
write $NS$ instead of $SN$ for one of the semi--elemental points in
order to have coherence of the positions of the parabolic (nodal)
sector of one point with respect to the hyperbolic (saddle) of the
other semi--elemental point. More concretely, the {\it Config. 3} in
Figure \ref{fig:top_dis_inf} must be described as
$\overline{\binom{1}{1}}SN,\,\overline{\binom{1}{1}}SN,\, N$ \label{page:SN} since the elemental node lies always
between the hyperbolic sectors of one saddle--node and the parabolic
ones of the other. However, the {\it Config. 4} in Figure
\ref{fig:top_dis_inf} must be described as
$\overline{\binom{1}{1}}SN$, $\overline{\binom{1}{1}} NS$, $N$
since the
hyperbolic sectors of each saddle--node lie between the elemental
node and the parabolic sectors of the other saddle--node. These two
configurations have exactly the same description of singular points
but their relative position produces topologically (and
geometrically) different portraits.

For the description of the topological phase portraits around the
isolated singular points of QS the information described above is
sufficient. However we are interested in additional geometric
features such as the number of characteristic directions which
figure in the final global picture of the desingularization. In
order to add this information we need to introduce more notation. If
two borsecs (the limiting orbits of a sector) arrive at the singular
point with the same direction, then the sector will be denoted by
$H_{\curlywedge}$, $E_{\curlywedge}$ or $P_{\curlywedge}$.
The index in this notation refers to the cusp--like form of limiting
trajectories of the sectors. Moreover, in the case of parabolic
sectors we want to make precise whether the orbits arrive tangent to
one borsec or to the other. We distinguish the two cases by
${\overset{ {}_\curvearrowleft}P}$ if they arrive tangent to the
borsec limiting the previous sector in clockwise sense or
${\overset{ {}_\curvearrowright}P}$ if they arrive tangent to the
borsec limiting the next sector. A parabolic sector will be $P^\ast$
when all orbits orbits arrive with all possible slopes between the
two consecutive borsecs. In the case of a cusp--like parabolic
sector, all orbits must arrive with only one direction, but the
distinction between ${\overset{ {}_\curvearrowleft}P}$ and
${\overset{ {}_\curvearrowright}P}$ is still valid if we consider
the different desingularizations we obtain from them. Thus,
complicated intricate singular points like the two we see in
Figure \ref{vicinities} may be described as $ \binom{4}{2}\overset{{}_\curvearrowright}PE\overset{
{}_\curvearrowleft}P-HHH $ (case $(a)$) and $ \binom{4}{3}
E {\overset{ {}_\curvearrowleft}P_{\curlywedge}}H-H
{\overset{ {}_\curvearrowright}P_{\curlywedge}}E $   (case
$(b)$), respectively.

Finally there is also the possibility that we have an infinite
number of infinite singular points.

\subsection*{Line at infinity filled up with singularities:}
 It is known that any such system has in a sufficiently small neighborhood of
infinity one of $6$ topological distinct phase portraits (see
\cite{Dana-Vlp-JDDE}). The way to determine these portraits is by
studying the reduced systems on the infinite local charts after
removing the degeneracy of the systems within these charts. In case
a singular point still remains on the line at infinity we study such
a point. In \cite{Dana-Vlp-JDDE} the tangential behavior of the
solution curves was not considered in the case of a node. If after
the removal of the degeneracy in the local charts at infinity a node
remains, this could either be of the type $N^d$, $N$ or $N^\star$
(this last case does not occur in quadratic systems as it was shown
in \cite{Art-Llib-Schl-Vlp-RMJM-2013}). Since no eigenvector of such a node $N$ (for
quadratic systems) will have the direction of the line at infinity
we do not need to distinguish $N^f$ and $N^\infty$. Other types of
singular points at infinity of quadratic systems, after removal of
the degeneracy, can be saddles, centers, semi--elemental
saddle--nodes or nilpotent elliptic--saddles. We also have the
possibility of no singularities after the removal of the degeneracy.
To convey the way these singularities were obtained as well as their
nature, we use the notation $[\infty;\,\emptyset]$, $[\infty;\,N]$,
$[\infty;\,N^d]$, $[\infty;\,S]$, $[\infty;\,C]$,
$[\infty;\,\overline{\binom{2}{0}} SN]$ or
$[\infty;\,\widehat{\binom{3}{0}} ES]$.


\section{Invariant polynomials and preliminary results} \label{sec:Invariant polyn.}

Consider real quadratic systems of the form
\begin{equation} \label{sys:QSgen}
\begin{gathered}
  \frac {dx}{dt}=p_0+ p_1(x,y)+\,p_2(x,y)\equiv P(x,y), \\
  \frac {dy}{dt}=q_0+ q_1(x,y)+\,q_2(x,y)\equiv Q(x,y),
\end{gathered}
 \end{equation}
with  homogeneous polynomials $p_i$ and $q_i$ $(i=0,1,2)$ of degree $i$
in $x,y$ written as
\begin{gather*}
 p_0=a_{00},\quad p_1(x,y)=  a_{10}x+ a_{01}y,\quad
p_2(x,y)= a_{20}x^2 +2 a_{11}xy + a_{02}y^2,\\
q_0=b_{00},\quad q_1(x,y)=  b_{10}x+ b_{01}y,\quad
q_2(x,y)= b_{20}x^2 +2 b_{11}xy + b_{02}y^2.
\end{gather*}
 Let $ \tilde a=(a_{00},a_{10},a_{01},a_{20},a_{11},a_{02},b_{00},
b_{10},b_{01},b_{20}, b_{11},b_{02})$ be the 12-tuple of the coefficients
of systems \eqref{sys:QSgen} and denote
$\mathbb{R}[\tilde  a,x,y]=\mathbb{R}[a_{00},\ldots, b_{02},x,y]$.


\subsection{Affine invariant polynomials associated with infinite singularities}

It is known that on the set \text{QS} of all quadratic differential
systems \eqref{sys:QSgen} acts the group $ Aff\,(2,\mathbb{R})$ of affine
transformations on the plane \mbox{(cf. \cite{Dana-Vlp-JDE})}. For
every  subgroup $G\subseteq  Aff\,(2,\mathbb{R})$ we have an  induced action
of $G$ on \text{QS}. We can identify the set \text{QS} of systems
\eqref{sys:QSgen} with a subset of $\mathbb{R}^{12}$ via the map
$\text{QS}\to \mathbb{R}^{12}$  which associates to each system
\eqref{sys:QSgen} the 12--tuple $(a_{00},\ldots,b_{02})$ of its
coefficients.

For the definitions of a $GL$--comitant  and invariant as well as
for the definitions of a $T$--comitant and a $CT$--comitant we
refer the reader to the paper \cite{Dana-Vlp-JDE} (see also
\cite{Sib1}). Here we shall only construct the necessary
$T$--comitants and $CT$--comitants associated to configurations of
singularities (including multiplicities)  of quadratic
systems
\eqref{sys:QSgen}.  All polynomials constructed here are $GL$--comitants.
But some are also affine invariants or even affine comitants.

Consider the polynomial
$\Phi_{\alpha,\beta}=\alpha P^*+\beta
Q^*\in \mathbb{R}[\tilde a,X,Y,Z,\alpha,\beta]$ where
$P^*=Z^2P(X/Z,Y/Z)$,
$Q^*=Z^2Q(X/Z,Y/Z)$, $P,Q\in \mathbb{R}[\tilde a,x,y]$ and
\[
\max\big(\deg_{(x,y)}P,\deg_{(x,y)}Q\big)=2.
\]
 Then
\begin{align*}
\Phi_{\alpha,\beta}
&= s_{11}(\tilde a,\alpha,\beta)X^2 +2
s_{12}(\tilde a,\alpha,\beta)XY+ s_{22}(\tilde a,\alpha,\beta)Y^2+
   2s_{13}(\tilde a,\alpha,\beta)XZ\\
&\quad +2s_{23}(\tilde a,\alpha,\beta)YZ+s_{33}(\tilde a,\alpha,\beta)Z^2
\end{align*}
and we denote
\begin{gather*}
 \widetilde D(\tilde a,x,y)
= 4 \det\|s_{ij}(\tilde a,y,-x)
    \|_{i,j\in\{1,2,3\}} ,\\
\widetilde H(\tilde a,x,y) = 4 \det\|s_{ij}(\tilde a,y,-x) \|_{i,j\in\{1,2\}}.
\end{gather*}

We consider the polynomials
\begin{equation} \label{expr:Ci,Dj}
\begin{gathered}
   C_i(\tilde a,x,y)=yp_i(\tilde a,x,y)-xq_i(\tilde a,x,y), \\
  D_i(\tilde a,x,y)=\frac{\partial}{\partial x}p_i(\tilde a,x,y)+
        \frac{\partial}{\partial y}q_i(\tilde a,x,y),
\end{gathered}
\end{equation}
in $\mathbb{R}[\tilde a,x,y]$ for $i=0,1,2$ and $i=1,2$ respectively. The
polynomials $C_2$  and $D_2$ are trivial $T$-comitants (trivial
because they only depend on the coefficients of the quadratic
terms).
Using the so--called  {\it transvectant of order $k$} (see {\rm
\cite{Gr_Yng}, \cite{Olver}}) of two polynomials $f,$ $g\in$
$\mathbb{R}[\tilde a,x,y]$
$$
  (f,g)^{(k)}=
   \sum_{h=0}^k (-1)^h \binom{k}{h}
   \frac{\partial^k f}{\partial x^{k-h}\partial y^h}\
   \frac{\partial^k g}{\partial x^h\partial y^{k-h}},
$$
we construct the following $GL$--comitants of the second degree
with respect to the coefficients of the initial system
\begin{equation} \label{expr:Ti}
\begin{gathered}
   T_1(\tilde a,x,y)=\left(C_0,C_1\right)^{(1)},\quad
   T_2(\tilde a,x,y)=\left(C_0,C_2\right)^{(1)},\quad
   T_{3}(\tilde a)=\left(C_0,D_2\right)^{(1)},\\
   T_{4}(\tilde a)=\left(C_1,C_1\right)^{(2)},\quad
   T_5(\tilde a,x,y)=\left(C_1,C_2\right)^{(1)},\quad
   T_6(\tilde a,x,y)=\left(C_1,C_2\right)^{(2)},\\
   T_7(\tilde a,x,y)=\left(C_1,D_2\right)^{(1)},\quad
   T_8(\tilde a,x,y)=\left(C_2,C_2\right)^{(2)},\quad
   T_9(\tilde a,x,y)=\left(C_2,D_2\right)^{(1)}.
\end{gathered}
\end{equation}
 Using these  $GL$--comitants as well as
the polynomials \eqref{expr:Ci,Dj} we construct the additional
invariant polynomials (see also \cite{Dana-Vlp-JDE})
\begin{gather*}
  \widetilde M(\tilde a,x,y)=  (C_2,C_2)^{(2)} \equiv 2
\operatorname{Hess}\big(C_2(\tilde a,x,y)\big);\\
    \eta(\tilde a)= (\widetilde M,\widetilde M)^{(2)}/384\equiv
\operatorname{Discrim}\big(C_2(\tilde a,x,y)\big);\\
   \widetilde K(\tilde a,x,y)=   \operatorname{Jacob}\big(p_2(\tilde a,x,y),
q_2(\tilde a,x,y)\big);\\
   K_1(\tilde a,x,y) =    p_1(\tilde a,x,y)q_2(\tilde a,x,y)
-p_2(\tilde a,x,y)q_1(\tilde a,x,y);\\
\begin{aligned}
  K_2(\tilde a,x,y) &=    4(T_2,\widetilde M-2\widetilde K)^{(1)}+3
     D_1(C_1,\widetilde M-2\widetilde K)^{(1)}\\
&\quad -(\widetilde M-2\widetilde  K)\big(16T_3-3T_4/2+3D_1^2\big);
\end{aligned}\\
   K_3(\tilde a,x,y) =   C_2^2(4T_3+3T_4)+C_2(3C_0\widetilde K-2C_1T_7)+
     2 K_1(3K_1-C_1D_2);\\
  \tilde L(\tilde a,x,y)=  4\widetilde K+8\widetilde H-\widetilde M;\\
    L_1(\tilde a,x,y)=  (C_2,\widetilde D)^{(2)};\\
    \widetilde R(\tilde a,x,y)=   \tilde L+8\widetilde K;\\
 \kappa(\tilde a)=  (\widetilde M,\widetilde K)^{(2)}/4; \\
 \kappa_1(\tilde a)=  (\widetilde M,C_1)^{(2)};\\
 \widetilde N(\tilde a,x,y)=   \widetilde K(\tilde a,x,y)
+ \widetilde H(\tilde a,x,y);
\\
\theta_2(\tilde{a})=(C_1,\widetilde{N})^{(2)}/16, \quad
\theta_6(\tilde a,x,y)=C_1 T_8-2C_2 T_6;
\\
\theta_5(\tilde{a})=2C_2(T_6,T_7)^{(1)} -(T_5+2D_2C_1)(C_1,D_2^2)^{(2)}.
\end{gather*}

The geometric meaning of the  invariant polynomials $C_2$,
$\widetilde M$ and $\eta$  is revealed in the next lemma (see
\cite{Dana-Vlp-JDE}).

\begin{lemma} \label{lem:S1-S5}
The form of the divisor $D_S(C,Z)$ for systems
\eqref{sys:QSgen} is determined by the corresponding conditions
indicated in Table 1, where we write $w_1^c+w_2^c+w_3$ if two of
the points, i.e. $w_1^c, w_2^c$, are complex but not real.
        Moreover, for each form of the divisor $D_S(C,Z)$ given
in Table 1 the quadratic systems \eqref{sys:QSgen} can be brought
via a linear transformation to one of the following  canonical
systems $(\textbf{S}_{I})-(\textbf{S}_{V})$ corresponding to the number and multiplicity at
infinity of their singularities at infinity.
\end{lemma}

\begin{table}[htb]%\label{table-1}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
  \raisebox{-0.7em}[0pt][0pt]{Case}
 & \raisebox{-0.7em}[0pt][0pt]{Form of $D_S(C,Z)$}
      & Necessary and    \\[-1mm]
         & & sufficient conditions  \\[-0.5mm]
         & & on the comitants \\
 \hline\hline
 \rule{0pt}{1.2em} $1$ & $w_1+w_2+w_3 $ &  $\eta>0 $ \\
\hline
 \rule{0pt}{1.2em}$2$  & $w_1^c+w_2^c+w_3 $ &  $\eta<0$ \\
\hline
 \rule{0pt}{1.2em}  $3$ & $2w_1+w_2 $ &  $\eta=0,\quad \widetilde M\ne0$ \\
\hline
 \rule{0pt}{1.2em} $4$ & $3w $ &  $ \widetilde M=0,\quad C_2\ne0$ \\
\hline
 \rule{0pt}{1.2em} $5$ & $D_S(C,Z)$ undefined  &  $ C_2=0$ \\
\hline
\end{tabular} \\ \quad \\
\textsc{Table} 1
\end{center}
\end{table}

%\label{sys:SI-SV}
\begin{gather*}
\begin{aligned}
 \dot x&=a+cx+dy+gx^2+(h-1)xy,\\
 \dot y&= b+ex+fy+(g-1)xy+hy^2;
\end{aligned} \quad (\textbf{S}_I)
\\
\begin{aligned}
 \dot x&=a+cx+dy+gx^2+(h+1)xy,\\
 \dot y&= b+ex+fy-x^2+gxy+hy^2;
\end{aligned}\quad  (\textbf{S}_{II})
\\
\begin{aligned}
 \dot x&=a+cx+dy+gx^2+hxy,\\[0.5mm]
 \dot y&=b+ex+fy+(g-1)xy+hy^2;
\end{aligned} \quad (\textbf{S}_{III})
\\
\begin{aligned}
 \dot x&=a+cx+dy+gx^2+hxy,\\
 \dot y&=b+ex+fy-x^2+gxy+hy^2,
\end{aligned} \quad (\textbf{S}_{IV})
\\
\begin{aligned}
 \dot x&=a+cx+dy+x^2,\\
 \dot y&= b+ex+fy+xy.
\end{aligned}\quad (\textbf{S}_{V})
\end{gather*}

\subsection{Affine invariant polynomials associated to  finite singularities}

Consider the differential operator ${\mathcal L}= x\cdot \mathbf{L}_2
-y\cdot\mathbf{L}_1$ acting on $\mathbb R[\tilde a,x,y]$ constructed in
\cite{Blt_Vlp_DEDS}, where
\begin{gather*}
   \mathbf{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}},\\
   \mathbf{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{gather*}
Using this operator and the affine invariant
$\mu_0={\rm Resultant}[p_2(\tilde a,x,1),q_2(\tilde a,x,1),x] $
we construct the following polynomials
\[
\mu_i(\tilde a,x,y) =\frac{1}{i!} {\mathcal L}^{(i)}(\mu_0), \quad
i=1,\dots,4,
\]
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:QSgen} with respect to the group $GL(2,\mathbb R)$ (see
\cite{Blt_Vlp_DEDS}). Their geometric meaning is revealed in
Lemmas \ref{lem:mu_i-Degen} and \ref{lem:mu_i-(0,0)} below.

\begin{lemma}[\cite{Blt_Vlp_BASM}] \label{lem:mu_i-Degen}
The total multiplicity of all finite singularities of a quadratic system
\eqref{sys:QSgen} equals $k$ if and only if for every
$i\in\{0,1,\ldots,k-1\}$ we have $\mu_i(\tilde a,x,y)=0$  in the ring
$\mathbb{R}[x,y]$   and $\mu_k(\tilde a,x,y)\ne0$. Moreover a system
\eqref{sys:QSgen} is degenerate (i.e. $\gcd(P,Q)\ne \text{constant}$) 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}


\begin{lemma}[\cite{Blt_Vlp_DEDS}] \label{lem:mu_i-(0,0)}
The point $M_0(0,0)$ is a singular point of multiplicity  $k$ ($1\le k
\le4$) for a quadratic system  \eqref{sys:QSgen} if and only if
for every $ i\in\left\{0,1,\ldots,k-1\right\}$ we have
$\mu_{4-i}(\tilde a,x,y)=0$ in $\mathbb R[x,y]$ and
$\mu_{4-k}(\tilde a,x,y)\ne0$.
\end{lemma}

We denote
$$
\sigma(\tilde a,x,y)=\frac{\partial  P}{\partial x}+\frac{\partial
Q}{\partial y}= \sigma_0(\tilde  a)+\sigma_1(\tilde  a,x,y)\ (\equiv
D_1(\tilde  a)+ D_2(\tilde  a,x,y)),
$$
and observe that  the polynomial  $\sigma(\tilde a,x,y)$ is an
affine comitant of systems \eqref{sys:QSgen}. It is known, that if
$(x_i,y_i)$ is a singular point of a system \eqref{sys:QSgen} then
for the trace of its respective linear matrix we have
$\rho_i =\sigma(x_i,y_i)$.

Applying the differential operators ${\mathcal L}$  and
$(*,*)^{(k)}$ (i.e. transvectant of index $k$) we  define the
following polynomial  function which governs the values of the
traces for finite singularities of systems \eqref{sys:QSgen}.

\begin{definition}[\cite{Vlp-NATMA}]\label{def:trase-F1} \rm
We call \textit{trace  polynomial} $\mathfrak{T} (w)$ over the ring
$\mathbb{R}[\tilde a]$ the polynomial defined as follows
\begin{equation} \label{fuct:T1st}
\mathfrak{T}
(w)=\sum_{i=0}^4\frac{1}{(i!)^2}\Big(\sigma_1^i,\,\frac{1}{i!}
{\mathcal L}^{(i)}(\mu_0)\Big)^{(i)} w^{4-i}=\sum_{i=0}^4
\mathcal{G}_i(\tilde a) w^{4-i},
\end{equation}
where the coefficients $\mathcal{G}_i(\tilde  a)=
\frac{1}{(i!)^2}(\sigma_1^i,\mu_i)^{(i)}\in\mathbb{R}[\tilde a]$,
 $i=0,1,2,3,4$ $\big(\mathcal{G}_0(\tilde  a)\equiv\mu_0(\tilde  a)\big)$
are $GL$--invariants.
\end{definition}

Using the polynomial $\mathfrak{T} (w)$ we could construct the
following four affine invariants $\mathcal{T}_4$,
$\mathcal{T}_3$, $\mathcal{T}_2$, $\mathcal{T}_1$, which are
responsible for the weak singularities:
$$
 { \mathcal{T}_{4-i}(\tilde a) } =\frac{1}{i!}
\frac{d^i  {\mathfrak{T} }}{dw^i}\Big|_{w=\sigma_0}\in\mathbb{R}[\tilde a], \quad
 i=0,1,2,3\quad  \big(\mathcal{T}_4\equiv \mathfrak{T}(\sigma_0)\big).
$$

The geometric meaning of these invariants is revealed by the next
lemma (see \cite{Vlp-NATMA}).

\begin{lemma} \label{lem:trases-formulas}
Consider a non-degenerate  system
\eqref{sys:QSgen}  and let $\mathbf{a}\in \mathbb{R}^{12}$ be its $12$-tuple of
coefficients. Denote by $\rho_s$   the trace of the linear part of
this system at a finite singular point $M_s$, $1\le s\le4$  (real or
complex, simple or multiple). Then the following relations hold.

\noindent (i) For  $\mu_0(\mathbf{a})\ne0$ (total multiplicity 4)
\begin{equation} \label{sim-poly:G0}
\begin{gathered}
\mathcal{T}_4(\mathbf{a})=\mathcal{G}_0(\mathbf{a})\rho_1\rho_2\rho_3\rho_4,\\
\mathcal{T}_3(\mathbf{a})=\mathcal{G}_0(\mathbf{a})(\rho_1\rho_2\rho_3+\rho_1\rho_2\rho_4
        +\rho_1 \rho_3\rho_4+ \rho_2\rho_3\rho_4),\\
\mathcal{T}_2(\mathbf{a})=\mathcal{G}_0(\mathbf{a})(\rho_1\rho_2+\rho_1\rho_3+\rho_1\rho_4
        +\rho_2 \rho_3+\rho_2\rho_4+ \rho_3\rho_4),\\
\mathcal{T}_1(\mathbf{a})=\mathcal{G}_0(\mathbf{a})(\rho_1+\rho_2+\rho_3+\rho_4).
\end{gathered}
\end{equation}

(ii) For  $\mu_0(\mathbf{a})=0$, $\mu_1(\mathbf{a},x,y)\ne0$ (total multiplicity 3)
\begin{equation} \label{sim-poly:G1}
\begin{gathered}
 \mathcal{T}_4(\mathbf{a})=\mathcal{G}_1(\mathbf{a})\rho_1\rho_2 \rho_3, \quad
 \mathcal{T}_3(\mathbf{a})=\mathcal{G}_1(\mathbf{a})(\rho_1\rho_2+\rho_1\rho_3+\rho_2\rho_3),\\
 \mathcal{T}_2(\mathbf{a})=\mathcal{G}_1(\mathbf{a})(\rho_1+\rho_2+\rho_3 ), \quad
\mathcal{T}_1(\mathbf{a})=\mathcal{G}_1(\mathbf{a}).
\end{gathered}
\end{equation}

(iii) For  $\mu_0(\mathbf{a})= \mu_1(\mathbf{a},x,y)=0$,
$\mu_2(\mathbf{a},x,y)\ne0$ (total multiplicity 2)
\begin{equation} \label{sim-poly:G2}
\begin{gathered}
 \mathcal{T}_4(\mathbf{a})=\mathcal{G}_2(\mathbf{a})\rho_1\rho_2,\quad
\mathcal{T}_3(\mathbf{a})=\mathcal{G}_2(\mathbf{a})(\rho_1+\rho_2),\\
 \mathcal{T}_2(\mathbf{a})=\mathcal{G}_2(\mathbf{a}),\quad
\mathcal{T}_1(\mathbf{a})=0.
\end{gathered}
\end{equation}

(iv) For  $\mu_0(\mathbf{a})= \mu_1(\mathbf{a},x,y)=\mu_2(\mathbf{a},x,y)=0$,
$\mu_3(\mathbf{a},x,y)\ne0$ (one elemental singularity)
\begin{equation} \label{sim-poly:G3}
\mathcal{T}_4(\mathbf{a})=\mathcal{G}_3(\mathbf{a})\rho_1,\quad
\mathcal{T}_3(\mathbf{a})=\mathcal{G}_3(\mathbf{a}),\quad
\mathcal{T}_2(\mathbf{a})= \mathcal{T}_1(\mathbf{a})=0.
\end{equation}
\end{lemma}

To  calculate the values of invariant
polynomials, we define here a family of $T$--comitants (see
\cite{Dana-Vlp-JDE} for detailed definitions)  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)
   -(C_1,T_5)^{(1)}\\
&\quad +6D_1(C_1D_2   -T_5)-9D_1^2C_2\Big]/36,
\end{aligned}
\\
\widehat  E=
\big[D_1(2T_9-T_8)-3\left(C_1,T_9\right)^{(1)}
      -D_2(3T_7+D_1D_2)\big]/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}+288 D_1\widehat  E\\
 &\quad -24(C_2,\widehat D)^{(2)}+120(D_2,\widehat D)^{(1)}
        -36C_1(D_2,T_7)^{(1)}+8D_1(D_2,T_5)^{(1)}\Big]/144,
\end{aligned}\\
\widehat K= (T_8+4T_9+4D_2^2)/72  \equiv   \widetilde K/4,\\
\widehat H= (8T_9-T_8+2D_2^2)/72 \equiv   -\widetilde H/4,\quad
\widehat  M= T_8, \\
\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
     -32D_1T_2+32\left(C_0, T_5\right)^{(1)}\big) \\
&\quad +6\left(D_2,    T_7\right)^{(1)}
 \big[8C_0(T_8-12T_9) -12C_1(D_1D_2+T_7)+ D_1(26C_2D_1+32T_5)\\
&\quad     + C_2(9T_{4}+96T_{3})\big] +6\left(D_2,
   T_6\right)^{(1)}\left[32C_0T_9-C_1(12T_7+52D_1D_2)
    -32C_2D_1^2\right]\\
&\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[D_1\left(C_1, T_6\right)^{(1)}  +D_2\left(C_0, T_6\right)^{(1)}]
 -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_{4}T_9\Big\} /(2^{8}3^{3}).
\end{aligned}
\end{gather*}
These polynomials in addition to \eqref{expr:Ci,Dj} and
\eqref{expr:Ti} will serve as bricks in constructing affine
invariant polynomials for   systems \eqref{sys:QSgen}.

The following 42 affine invariants   $A_1,\ldots,A_{42}$ form the
minimal polynomial basis of affine invariants up to degree 12.
This fact was proved in \cite{Bul-Tim} by  constructing
$A_1,\ldots,A_{42}$ using the above  bricks.
\begin{alignat*}2
& A_1 = \hat   A,
&&
\begin{aligned}
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)},
\end{aligned}\\
& A_2 = (C_2, \widehat   D)^{(3)}/12,
&& A_{23} = \big[\widehat  F, \widehat H)^{(1)}, \widehat K\big)^{(2)}/8,\\
& A_{3} =  \big[ C_2, D_2)^{(1)}, D_2\big)^{(1)},
D_2\big)^{(1)}/48, \quad
&& A_{24} = \big[C_2, \widehat  D)^{(2)}, \widehat K\big)^{(1)}, \widehat H\big)^{(2)}/32,\\
& A_{4} = (\widehat H, \widehat H)^{(2)},
&& A_{25} = \big[\widehat  D, \widehat  D)^{(2)}, \widehat  E\big)^{(2)}/16,\\
& A_5 =  (\widehat H, \widehat K)^{(2)}/2,
&& A_{26} = (\widehat  B, \widehat  D)^{(3)}/36,\\
& A_6 =  (\widehat  E, \widehat H)^{(2)}/2,
&& A_{27} = \big[\widehat  B, D_2)^{(1)}, \widehat  H\big)^{(2)}/24,\\
& A_7 = \big[ C_2, \widehat  E)^{(2)}, D_2\big)^{(1)}/8,
&& A_{28} = \big[C_2,\widehat K)^{(2)},\widehat  D\big)^{(1)},\widehat  E\big)^{(2)}/16,\\
& A_8 = \big[\widehat  D, \widehat  H)^{(2)},
D_2\big)^{(1)}/8,
&& A_{29} = \big[\widehat  D, \widehat  F)^{(1)}, \widehat  D\big)^{(3)}/96,\\
& A_9   = \big[\widehat  D, D_2)^{(1)}, D_2\big)^{(1)},
D_2\big)^{(1)}/48,
&& A_{30} = \big[C_2,\widehat  D)^{(2)},\widehat  D\big)^{(1)},\widehat  D\big)^{(3)}/288,\\
& A_{10} = \big[\widehat  D, \widehat K)^{(2)},
D_2\big)^{(1)}/8,
&& A_{31} = \big[\widehat  D,\widehat  D)^{(2)},\widehat K\big)^{(1)},\widehat  H\big)^{(2)}/64,\\
& A_{11} = (\widehat  F, \widehat K)^{(2)}/4,
&& A_{32} = \big[\widehat  D, \widehat  D)^{(2)}, D_2\big)^{(1)}, \widehat  H\big)^{(1)}, D_2\big)^{(1)}/64,\\
& A_{12} = (\widehat  F, \widehat  H)^{(2)}/4,
&& A_{33} = \big[\widehat  D, D_2)^{(1)}, \widehat  F\big)^{(1)}, D_2\big)^{(1)}, D_2\big)^{(1)}/128,\\
& A_{13} = \big[C_2, \widehat  H)^{(1)}, \widehat H\big)^{(2)},
D_2\big)^{(1)}/24,
&& A_{34} = \big[\widehat  D, \widehat  D)^{(2)}, D_2\big)^{(1)}, \widehat K\big)^{(1)}, D_2\big)^{(1)}/64,\\
& A_{14} = (\widehat  B, C_2)^{(3)}/36,
&& A_{35} = \big[\widehat  D, \widehat  D)^{(2)}, \widehat  E\big)^{(1)}, D_2\big)^{(1)}, D_2\big)^{(1)}/128,\\
& A_{15} = (\widehat  E, \widehat  F)^{(2)}/4,
&& A_{36} = \big[\widehat  D,\widehat  E)^{(2)},\widehat  D\big)^{(1)},\widehat  H\big)^{(2)}/16,\\
& A_{16} = \big[\widehat  E, D_2)^{(1)}, C_2\big)^{(1)},
\widehat K\big)^{(2)}/16,
&& A_{37} = \big[\widehat  D,\widehat  D)^{(2)},\widehat  D\big)^{(1)},\widehat  D\big)^{(3)}/576,\\
& A_{17} = \big[\widehat  D,\widehat
D)^{(2)},D_2\big)^{(1)},D_2\big)^{(1)}/64,
&& A_{38} = \big[C_2,\widehat  D)^{(2)}, \widehat  D\big)^{(2)}, \widehat  D\big)^{(1)}, \widehat  H\big)^{(2)}/64,\\
& A_{18} = \big[\widehat  D,\widehat  F)^{(2)},
D_2\big)^{(1)}/16,
&& A_{39} = \big[\widehat  D,\widehat  D)^{(2)},\widehat  F\big)^{(1)},\widehat  H\big)^{(2)}/64,\\
& A_{19} = \big[\widehat  D,\widehat  D)^{(2)},\widehat
H\big)^{(2)}/16,
&& A_{40} = \big[\widehat  D,\widehat  D)^{(2)},\widehat  F\big)^{(1)},\widehat K\big)^{(2)}/64,\\
& A_{20} = \big[C_2,\widehat  D)^{(2)}, \widehat
F\big)^{(2)}/16,
&& A_{41} = \big[C_2,\widehat  D)^{(2)}, \widehat  D\big)^{(2)}, \widehat  F\big)^{(1)}, D_2\big)^{(1)}/64,\\
& A_{21} = \big[\widehat  D, \widehat  D)^{(2)}, \widehat
K\big)^{(2)}/16,
&& A_{42} = \big[\widehat  D,\widehat  F)^{(2)},\widehat  F\big)^{(1)},D_2\big)^{(1)}/16.\\
\end{alignat*}

In the above list, the bracket ``$[$'' is used in order to avoid
placing the otherwise necessary up to five parenthesizes ``$($''.

Using the elements of the minimal polynomial basis given above we
construct the  affine invariants
\begin{gather*}
\mathcal{F}_1(\tilde a)=A_2,\\
\begin{aligned}
\mathcal{F}_2(\tilde a)
&=-2 A_1^2 A_{3} + 2 A_5 (5 A_8 + 3 A_9) +
 A_{3} (A_8 - 3 A_{10} + 3 A_{11} + A_{12}) \\
 &\quad - A_{4} (10 A_8 - 3 A_9 + 5 A_{10} + 5A_{11} + 5A_{12}),
\end{aligned} \\
\begin{aligned}
\mathcal{F}_3(\tilde a)&= -10 A_1^2 A_{3} + 2 A_5 (A_8 - A_9) -
 A_{4} (2 A_8 + A_9 + A_{10} + A_{11} + A_{12}) +  \\
&\quad + A_{3} (5 A_8 + A_{10} - A_{11} + 5 A_{12}),
\end{aligned}\\
\begin{aligned}
\mathcal{F}_4(\tilde a)
&= 20 A_1^2 A_2 - A_2 (7 A_8 - 4 A_9 + A_{10} + A_{11} + 7 A_{12}) +
 A_1 (6 A_{14} - 22 A_{15}) \\
&\quad -  4 A_{33} + 4 A_{34},
\end{aligned} \\
\mathcal{F}(\tilde a)= A_7,\\
\mathcal{B}(\tilde a)= -(3 A_8 + 2 A_9 + A_{10} + A_{11} + A_{12}),\\
\mathcal{H}(\tilde a)=-(A_{4}+2A_5),
\end{gather*}
as well as the   $GL$-comitants,
\begin{gather*}
\begin{aligned}
\mathcal{B}_1(\tilde a)
&=\Big\{\big(T_7,D_2\big)^{(1)}\big[12 D_1 T_3+2D_1^3+9D_1T_4
 +36\big(T_1,D_2\big)^{(1)}\big]
  - 2D_1\big(T_6,D_2\big)^{(1)}\\
&\quad\times \big[D_1^2 +12T_3]+D_1^2 \big[D_1\big(T_8,C_1\big)^{(2)}+
           6\big(\big(T_6,C_1\big)^{(1)},D_2\big)^{(1)}\big]\Big\}/144,
\end{aligned}\\
\begin{aligned}
\mathcal{B}_2(\tilde a)
&=\Big\{\big(T_7,D_2\big)^{(1)}\big[8T_3\big(T_6,D_2\big)^{(1)}-
D_1^2\big(T_8,C_1\big)^{(2)}-
          4D_1\big(\big(T_6,C_1\big)^{(1)},D_2\big)^{(1)}\big]\\
&\quad +[\big(T_7,D_2\big)^{(1)}]^2(8T_3-3T_4+2D_1^2)\Big\}/384,
\end{aligned}\\
\mathcal{B}_{3}(\tilde a,x,y)
= -D_1^2(4D_2^2+T_8+4T_9)+3D_1D_2(T_6+4T_7)-24T_3(D_2^2-T_9),\\
\mathcal{B}_{4}(\tilde a,x,y)= D_1(T_5+2D_2C_1)-3C_2(D_1^2+2T_3).
\end{gather*}
We note that the invariant polynomials $\mathcal{T}_i$,
$\mathcal{F}_i$, $\mathcal{B}_i$\ (i=1,2,3,4), and $\mathcal{B}$,
$\mathcal{F}$, $\mathcal{H}$ and $\sigma$  are responsible for
weak singularities of the family of quadratic systems (see
\cite[Main Theorem]{Vlp-NATMA}).

Now we need also the invariant polynomials which are responsible
for the types of the finite singularities. These  were constructed
in  \cite{Art-Llib-Vlp-IJBCh}. Here we need only the following
ones (we keep the   notation from \cite{Art-Llib-Vlp-IJBCh}):
%\label{inv:W_i-Appendix}
\begin{align*}
W_3(\tilde a) &= \big[9 A_1^2(36 A_{18} - 19 A_2^2 + 134 A_{17}  + 165 A_{19})
   +3 A_{11}(42 A_{18} -102 A_{17} \\
&\quad +195 A_{19})  +2 A_2^2( A_{10}+3 A_{11})+102 A_3(3A_{30} -14    A_{29})\\
&\quad  -63 A_6(17 A_{25}+ 30 A_{26}) +3 A_{10}(14 A_{18}
 -118 A_{17}+ 153 A_{19} +120 A_{21})\\
&\quad +6 A_7( 329 A_{25}  - 108 A_{26}) +3 A_8(164 A_{18}
  + 153 A_{19} -442 A_{17}) \\
&\quad +9 A_{12}(2 A_{20} -160 A_{17} -2 A_{18} -59 A_{19})
          +3 A_1(77 A_2A_{14}\\
&\quad +235 A_2A_{15} - 54 A_{36})   +18 A_{21}(21 A_9 -5 A_{11}) + 302 A_2A_{34}
         -366 A_{14}^2 \\
&\quad -12 A_{15}(71 A_{14} + 80 A_{15})\big]/9,
\end{align*}
\begin{align*}
&W_4(\tilde a) \\
&= \big[1512A_1^2(A_{30} - 2A_{29})-648 A_{15}A_{26}
       +72 A_1A_2(49 A_{25}+ 39 A_{26})\\
&\quad +6A_2^2(23 A_{21} - 1093 A_{19})
       -87 A_2^4  +4 A_2^2(61 A_{17} + 52 A_{18} + 11 A_{20})\\
&\quad -6A_{37}(352 A_{3} + 939 A_{4} - 1578 A_5)
       -36A_8(396 A_{29} +265 A_{30})\\
&\quad +72A_{29}(17 A_{12} -38 A_9 -109 A_{11})
       +12 A_{30}(76 A_9 -189 A_{10} - 273 A_{11} -651 A_{12})\\
&\quad -648 A_{14}(23 A_{25}+ 5 A_{26})
        -24 A_{18}(3 A_{20} + 31 A_{17})
       +36 A_{19}(63 A_{20} + 478 A_{21})\\
&\quad  +18 A_{21}(2 A_{20} +137 A_{21})
         -4A_{17}(158 A_{17} + 30 A_{20} +87 A_{21})\\
&\quad -18 A_{19}(238 A_{17} + 669   A_{19})\big]/81,
\end{align*}
\begin{align*}
W_7(\tilde a) &=   12A_{26}(A_{26}-2A_{25})+(2A_{29}-A_{30})(A_2^2
         -20A_{17}-12A_{18}+6A_{19}+6A_{21})\\
&\quad  +48A_{37}(A_1^2-A_8-A_{12}),
\end{align*}
\begin{align*}
W_8(\tilde a)
&=  64D_1\big[\big(\big(T_6,C_1\big)^{(1)},D_2\big)^{(1)}\big]^2
                  \big[16 \big(C_0,T_6\big)^{(1)}
             -37\big(D_2,T_1\big)^{(1)}+12D_1T_3\big] \\
&\quad +4(108D_1^4-3T_4^2  -128T_3T_4+42D_1^2T_4)
 \big[\big(\big(T_6,C_1\big)^{(1)},D_2\big)^{(1)}\big]^2
\\
&\quad +36D_1\big(\big(T_6,C_1\big)^{(1)},D_2\big)^{(1)}
 \big[4D_1\big(C_0,T_6\big)^{(1)}
  -D_1^2(4T_3+T_4) +24T_3^2\big]\\
&\quad\times \big(C_1,T_8\big)^{(2)}
  + 64\big[\big(\big(T_6,C_1\big)^{(1)},D_2\big)^{(1)}\big]^2
  \big[27T_3^2
 +16\big(\big(T_6,C_1\big)^{(1)},C_0\big)^{(1)}\big]  \\
&\quad-54 \big[8D_1^4+D_1^2T_4  -8  D_1\big(C_0,T_6\big)^{(1)}
  +8D_1^2T_3 +8T_3^2\big]
\big(\big(T_6,C_1\big)^{(1)},T_6\big)^{(1)}\\
 &\quad\times   \big(C_1,T_8\big)^{(2)}
 + 108D_1T_3\big[\big(C_1,T_8\big)^{(2)}\big]^2
                 \big[D_1T_3-2\big( C_0,T_6\big)^{(1)}\big]\\
&\quad +576\big(\big(T_6,C_1\big)^{(1)},D_2\big)^{(1)}
                  \big(\big(T_6,C_1\big)^{(1)},T_6\big)^{(1)}
               \big[2\big(D_2,T_1\big)^{(1)}-5D_1T_3\big]\\
&\quad -27\big[\big(C_1,T_8\big)^{(2)}\big]^2
                 \big[T_4^4/8 + \big( C_0,T_1\big)^{(1)}\big],
\end{align*}
\begin{gather*}
W_9(\tilde a)  = -3T_4(T_4 +4D_1^2)- 32D_1\left(C_0,T_6\right)^{(1)}
               -32 \left(\left(T_6,C_1\right)^{(1)},C_0\right)^{(1)},\\
W_{11}(\tilde a,x,y)= 2\widetilde K(C_2,\widetilde D)^{(2)}+12
\widetilde F(2\widetilde H+\widetilde K)-D_2(\widetilde
D,\widetilde K)^{(1)},
\\
F_4(\tilde a,x,y) = \mu_3(\tilde a,x,y), \quad
F_5(\tilde a,x,y) = T_5+2C_1D_2-3C_2D_1,\\
G_3(\tilde a) = A_2, \quad
\mathbf{D}(\tilde a) = \big[3((\mu_3,\mu_3)^{(2)},\mu_2)^{(2)}
-(6\mu_0\mu_4-3\mu_1\mu_3+\mu_2^2,\mu_4)^{(4)}\big]/48,\\
G_8(\tilde a) = (5A_8-18A_1^2-A_{10}-3A_{11}+9A_{12})/2,\quad
\mathbf{U}(\tilde a,x,y)=\mu_3^2-4\mu_2\mu_4.
\end{gather*}

Finally we need the invariant polynomials which are responsible for
the existence of one (or two) star node(s) arbitrarily located on
the phase plane of a system  \eqref{sys:QSgen}. We have the
following lemma (see \cite{Vul-Lup-BASM}).

\begin{lemma} \label{lem:Star-nodes}
 A quadratic system \eqref{sys:QSgen}
possesses  one star node     if and only if  one of the following
set of conditions holds:
\begin{itemize}
\item[(i)]  $U_1\ne0$,    $U_2\ne0$,  $U_3=Y_1= 0$;
\item[(ii)] $U_1=U_4=U_5=U_6=0$,  $Y_2\ne0$;
\end{itemize}
 and it possesses two   star nodes   if and only if
\begin{itemize}
\item[(iii)] $U_1=U_4=U_5=0$, $U_6\ne0$, $Y_2>0$,
 where
\begin{gather*}
  U_1(\tilde a,x,y) = \widetilde N ,\quad
 U_2 (\tilde a,x,y)  =  (C_1, \widetilde H-\widetilde K)^{(1)}  - 2D_1 \widetilde N,\\
 U_3(\tilde a,x,y) = 3 \widetilde D (D_2^2 - 16 \widetilde K)+
   C_2\big[(C_2, \widetilde D)^{(2)} - 5 (D_2, \widetilde D)^{(1)} +
   6\, \widetilde F \big],\\
 U_4(\tilde a,x,y)= 2T_5  +C_1 D_2,\quad
 U_5(\tilde a,x,y)= 3 C_1 D_1+4T_2-2C_0 D_1,\\
  U_6(\tilde a,x,y)= \widetilde H,\quad
 Y_1(\tilde a)= A_1,\quad
 Y_2(\tilde a,x,y)=2D_1^2+8T_3-T_4.
\end{gather*}
\end{itemize}
\end{lemma}


We base our work here on the results obtained in
\cite{Art-Llib-Schl-Vlp-RMJM-2013,Art-Llib-Vlp-IJBCh,Vlp-NATMA}.

\section{Proof of the main theorem}\label{sec:the proof}

 According to
\cite{Vlp-NATMA} for the quadratic systems having the finite
singularities of total multiplicity 2   the conditions
$\mu_0=\mu_1=0$ and $\mu_2\ne0$  must be satisfied. So by
\cite{Art-Llib-Schl-Vlp-RMJM-2013} the following lemma is valid.

\begin{lemma} \label{lem:ISP-mf=2}
The configurations of singularities at
infinity of the family of quadratic systems possessing finite
singularities (real or complex) of total multiplicity $2$ (i.e.
$\mu_0=\mu_1=0$ and $\mu_2\ne0$) are classified in
Diagram \ref{diagram:mf=2-ISP} according to the geometric
equivalence relation. Necessary and sufficient conditions for each
one of the $43$ different equivalence classes can be assembled from
these diagrams in terms of $14$ invariant polynomials with respect
to the action of the affine group and time rescaling, given in
Section~\ref{sec:Invariant polyn.}.
\end{lemma}

\begin{table}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth, height=0.8\textheight]{diagram4a}
% Diagr-mf2-a2.EPS
\end{center}
 \caption{The case $\mu_0=\mu_1=0$, $\mu_2\ne 0$.}\label{diagram:mf=2-ISP}
\end{table}

\begin{table}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{diagram4b}
% Diagr-mf2-b2.EPS
\\
\textsc{Diagram} \ref{diagram:mf=2-ISP} ({\it continued}).
The case $\mu_0=\mu_1=0$, $\mu_2\ne 0$.
\end{center}
\end{table}


\subsection{The family of quadratic differential systems with only two
distinct complex  finite singularities}
 Assuming that
quadratic systems \eqref{sys:QSgen}   possess two finite complex
singular points, according to \cite{Vlp-NATMA}  (see Table 2) we
have to consider two cases: $\widetilde K\ne0$ and  $\widetilde
K=0$.


\subsubsection{Systems with $\widetilde K\ne0$.} In this case
according to \cite{Vlp-NATMA} we shall consider the following
family of systems
\begin{equation} \label{sys:15a}
\dot x= a + h u x + 2 h x y + a y^2,\quad
\dot y = b + m u x + 2 m x y + b y^2,
\end{equation}
possessing the singular points $M_{1,2}(0,\pm i)$. For these
systems calculations yield
\begin{equation} \label{val:15a}
\begin{gathered}
\mu_0=\mu_1=0,\quad \mu_2= (b h - a m)^2 (4 + u^2) y^2,\quad
\widetilde K=4 (b h - a m) y^2,\\
\kappa= -128 m^2 (b h - a m),\quad
 \eta=4 m^2  \big[(b + 2 h)^2 - 8 (b h - a m)\big],\\
 \widetilde M=-32 m^2 x^2 - 16 (b - 2 h) m x y - 8\big[ (b - 2 h)^2 + 6 a m\big]
 y^2.
\end{gathered}
\end{equation}

\begin{remark} \label{rem:kappa>0->eta>0} \rm
We observe that  $\mu_2>0$ and if $\kappa\ne0$ then $ \widetilde
M\ne0$. Moreover the condition $\kappa>0$ implies $\eta>0$.
\end{remark}

\begin{remark} \label{rem:15a-m=1,0} \rm
The family of systems \eqref{sys:15a}
depends on five parameters. However due to a rescaling we can reduce
the number of the parameters to three. More precisely since by the
condition $\widetilde K\ne0$ (i.e. $b h - a m\ne0$) we have
$m^2+h^2\ne0$, then we may assume $(m,h)\in\{(1,1), (1,0), (0,1)\}$
due to the rescaling: $(i)$ $(x,y,t)\mapsto(hx/m, y,t/h)$ if
$mh\ne0$; $(ii)$ $(x,y,t)\mapsto(x/m, y,t)$ if $h=0$, and $(iii)$
$(x,y,t)\mapsto(x, y,t/h)$ if $m=0$.
 \end{remark}

Considering \eqref{val:15a} and $\widetilde K\ne0$ we deduce that the condition
$m\ne0$ is equivalent to $\kappa\ne0$.

\subsubsection*{The case $\kappa\ne0$.} Then considering  Remark
\ref{rem:15a-m=1,0} we shall examine the subfamilies of systems
\eqref{sys:15a} with  $(m,h)=(1,1)$  and  $(m,h)=(1,0)$.

\subsection*{A. Systems with $m=h=1$.} We consider the
3-parameter family of systems
\begin{equation} \label{sys:15a)-m=h=1}
\dot x= a +   u x + 2   x y + a y^2,\quad
\dot y  = b +   u x + 2 x y + b y^2,
\end{equation}
for which calculations yield
\begin{equation} \label{val:15a)-m=h=1}
\begin{gathered}
\mu_0=\mu_1=0,\quad \mu_2= (a - b)^2 (4 + u^2) y^2 ,\quad
 \widetilde K= 4 (b-a) y^2,\\ \kappa= 128 (a - b), \\
 \eta=4 \big[(b-2)^2 + 8 a\big],\quad \theta=64(b-a).
\end{gathered}
\end{equation}

\subsubsection*{The subcase $\kappa<0$.} Since $\mu_2>0$ and
$\widetilde M\ne0$ (see Remark \ref{rem:kappa>0->eta>0}) according to
Lemma \ref{lem:ISP-mf=2} we get the following three global
configurations of singularities:
\begin{gather*}
{\scriptstyle\copyright,\copyright};\,
\overline{\binom{2}{1}} N, \copyright,\copyright:
\text{ Example } \Rightarrow (a=-1,\; b= 0,\; u=0) \quad (\text{if }\eta<0);
\\
 {\scriptstyle\copyright,\copyright};\,
\overline{\binom{2}{1}} N, S, N^\infty : \text{ Example }
\Rightarrow (a=0,\; b= 1,\; u=0) \quad (\text{if }\eta>0);
\\
{\scriptstyle\copyright,\copyright};\,
\overline{\binom{0}{2}} SN,\overline{\binom{2}{1}} N
: \text{ Example } \Rightarrow (a=0,\; b= 2,\; u=0) \quad (\text{if }\eta=0).
\end{gather*}

\subsubsection*{The subcase $\kappa>0$.} By Remark
\ref{rem:kappa>0->eta>0} we have $\eta>0$ and   considering Lemma
\ref{lem:ISP-mf=2} we arrive at the global configuration of
singularities
\[
 {\scriptstyle\copyright,\copyright};\,
\overline{\binom{2}{1}}S, N^f, N^f : \text{ Example }
\Rightarrow (a=1,\; b= 0,\; u=0).
\]

\subsection*{B. Systems with $m=1,h=0$.} We consider the
3-parameter family of systems
\begin{equation} \label{sys:15a)-m=1,h=0}
\dot x= a +     a y^2,\quad
\dot y  = b +   u x + 2   x y + b y^2,
\end{equation}
where we may assume $b\in\{0,1\}$ due to the rescaling
$(x,y,t)\mapsto(bx,\, y,\,  t/b)$ (if $b\ne0$). For these systems
calculations yield
\begin{equation} \label{val:15a)-m=1,h=0}
\mu_0=\mu_1=0,\quad \mu_2= a^2 (4 + u^2) y^2 ,\quad
 \widetilde K= -4 a y^2,\quad \kappa= 128 a, \quad
 \eta= 4 (8 a + b^2).
\end{equation}
Considering Remark \ref{rem:kappa>0->eta>0}   we conclude that the
above systems could not possess new configurations  different
from the configurations of  systems \eqref{sys:15a)-m=h=1}.


\subsubsection*{The case $\kappa=0$.}
Considering \eqref{val:15a}, due to $\widetilde K\ne0$ we obtain
$m=0$ and then by Remark \ref{rem:15a-m=1,0} we may assume $h=1$.
Thus we arrive at the following systems
\begin{equation} \label{sys:15a)-m=0,h=1}
\dot x= a +  u x + 2  x y + a y^2,\quad \dot y  = b   + b y^2,
\end{equation}
where due to the rescaling $(x,y,t)\mapsto(ax,  y, t)$
(if $a\ne0$) we can assume $a\in\{0,1\}$.

For these systems we calculate
\begin{equation} \label{val:15a)-m=0}
\begin{gathered}
\mu_0=\mu_1=\kappa=\kappa_1=0,\quad \mu_2= b^2 (4 + u^2) y^2,\quad
 \widetilde K=4 b y^2, \\
 \tilde L=8 b(b-2) y^2,\quad   K_2=-384 b^2 (4 - 3 b + b^2) y^2,\\
\eta=0,\quad \widetilde M=-8 (b-2)^2 y^2,\quad
 C_2=(2 - b) x y^2 + a y^3.\\
\end{gathered}
\end{equation}

\subsubsection*{The subcase $\widetilde K<0$.} Then $b<0$ and this
implies $\widetilde M\ne0$. We observe that $K_2<0$, $\mu_2>0$ and
considering Lemma \ref{lem:ISP-mf=2} we obtain the configuration
\[
 {\scriptstyle\copyright,\copyright};\,\binom{2}{2}\,
H- H,\,N^f : \text{ Example } \Rightarrow (a=0,\; b= -2,\; u=0).
\]

\subsubsection*{The subcase $\widetilde K>0$.} We consider two
possibilities: $\tilde L\ne0$ and $\tilde L=0$.
\smallskip

 (1) {\it The possibility $\tilde L\ne0$.} In
this case $\widetilde M\ne0$ and taking into account the conditions
$K_2<0$ and $\mu_2>0$ by Lemma \ref{lem:ISP-mf=2} we arrive at the
following two  global configurations of singularities
\begin{gather*}
 {\scriptstyle\copyright,\copyright};\, \binom{2}{2}E-E,\,S
 : \text{ Example } \Rightarrow (a=0,\; b= 1,\; u=0) \quad (\text{if }\tilde L<0);
\\
 {\scriptstyle\copyright,\copyright};\, \binom{2}{2}H-H,\,N^\infty : \text{ Example } \Rightarrow (a=0,\;
b=3,\; u=0) \quad (\text{if }\tilde L>0).
\end{gather*}

(2) {\it The possibility $\tilde L=0$.}  Since $b\ne0$ (due
to $\mu_2\ne0$) we have $b=2$ and then $\widetilde M=0$ and $C_2=a
y^3$.  So considering   Lemma \ref{lem:ISP-mf=2} we obtain the
following two configurations
\begin{gather*}
 {\scriptstyle\copyright,\copyright};\, \binom{2}{3}
\overset{{}_\curvearrowright}P-\overset{
{}_\curvearrowleft}P : \text{ Example } \Rightarrow (a=1,\;
b= 2,\; u=0) \quad (\text{if }C_2\ne0);
\\
 {\scriptstyle\copyright,\copyright};\, [\infty;\, C]
: \text{ Example } \Rightarrow (a=0,\; b= 2,\; u=0) \quad (\text{if }C_2=0).
\end{gather*}

\subsubsection{Systems with $\widetilde K=0$.} In this case
according to \cite{Vlp-NATMA} we consider the following
family of systems
\begin{equation} \label{sys:15b}
\dot x=a + c x + g x^2 + 2 h x y + a y^2,\quad \dot y = x,\quad
(a\ne0)
\end{equation}
for which we calculate
\begin{equation} \label{val:15b}
\begin{gathered}
\mu_0=\mu_1=\kappa=0,\quad \mu_2= a (g x^2 + 2 h x y + a y^2)x^2 ,\quad
\widetilde K=0, \\
\tilde L=8 g (g x^2 + 2 h x y + a y^2),\quad  \eta=  4 g^2 (h^2-a g),\\
\theta_2=h^2-a g,\quad \widetilde M=-8 g^2 x^2 - 16 g h x y + 8 (3 a g - 4 h^2) y^2.
\end{gathered}
\end{equation}
As $\mu_2\ne0$ we have $\operatorname{sign}(\mu_2\tilde L)=\operatorname{sign}(ag)$.

\subsubsection*{The case $\eta<0$.} Then $\theta_2\ne0$ and considering
Lemma \ref{lem:ISP-mf=2} we arrive at the configuration
\[
 {\scriptstyle\copyright,\copyright};\,
N^d,\,\binom{1}{1}\,\copyright,\binom{1}{1}\, \copyright
 : \text{ Example } \Rightarrow (a=2,\;c=0,\; g= 1,\; h=1).
\]

\subsubsection*{The case $\eta>0$.} As $\theta_2\ne0$   considering
Lemma \ref{lem:ISP-mf=2} we obtain the following two
 global configurations of singularities
\begin{gather*}
\begin{aligned}
 {\scriptstyle\copyright,\copyright};\,
\overline{\binom{1}{1}} SN,\,\overline{\binom{1}{1}}
SN,\,N^d : \text{ Example } \Rightarrow &(a=1,\; c=0,\; g=
-1,\; h=1)\\
 & (\text{if }\mu_2\tilde L<0);
\end{aligned}\\
\begin{aligned}
 {\scriptstyle\copyright,\copyright};\,
\overline{\binom{1}{1}}SN,\,\overline{\binom{1}{1}}
NS,\,N^d : \text{ Example } \Rightarrow &(a=1,\; c=0,\; g=
1/2,\; h=1) \\
&(\text{if }\mu_2\tilde L>0).
\end{aligned}
\end{gather*}

\subsubsection*{The case $\eta=0$.}

\subsubsection*{The subcase $\tilde L\ne0$.} Then $g\ne0$ and we
obtain $h^2-ag=0$ and as  $a\ne0$ we get $g=h^2/a$. Calculations
yield
\begin{gather*}
\tilde L= 8 h^2 (h x + a y)^2 /a^2,\quad \widetilde M=- 8 h^2 (h
x + a y)^2 /a^2,\\
\kappa_1= 32 h^2 (a + c h) /a,\quad
 \theta_5=- 96 h (a + c h) (h x + a y)^3 /a^2.
\end{gather*}
As we observe the condition $\tilde L\ne0$ implies
$\widetilde M\ne0$, i.e. at infinity we have two distinct singularities.
\smallskip

(1) {\it The possibility $\kappa_1\ne0$.} Then $\theta_5\ne0$
and  considering Lemma \ref{lem:ISP-mf=2} we obtain the
configuration
\[
 {\scriptstyle\copyright,\copyright};\,
\widehat{\binom{2}{2}} {\overset{
  {}_\curvearrowright}P_{\curlywedge}}\overset{
  {}_\curvearrowleft}P  {H_{ \curlywedge}}-H,\,N^d :
\text{ Example } \Rightarrow (a=1,\;c=0,\; g= 1,\; h=1).
\]

(2) {\it The possibility $\kappa_1=0$.} As $\tilde L\ne0$ we get
$a=-ch\ne0$ and then we have
$$
K_2=- 384 h^4 (x - c y)^2/c^2,\quad \theta_6= 8 h^2 ( x - c y)^4
/c^2,\quad \mu_2=h^2 x^2 (x - c y)^2\ne0.
$$
So we obtain $K_2<0$, $\theta_6\ne0$ and considering Lemma
\ref{lem:ISP-mf=2} we obtain the configuration  of singularities
\[
 {\scriptstyle\copyright,\copyright};\, \binom{2}{2} H-H,\,N^d: \text{ Example } \Rightarrow (a=-1,\; c=1,\; g= -1,\; h=1).
\]

\subsubsection*{The subcase $\tilde L=0$.} In this case
considering \eqref{val:15b} we get $g=0$ and then we calculate
\begin{gather*}
\eta=\tilde L= 0,\quad  \widetilde M=-32 h^2 y^2,\quad
 C_2=y^2 (2 h x + a y),\\
\kappa_1= 128 h^2,\quad \mu_2=a y (2 h x + a y)\ne0.
\end{gather*}
We observe that the condition $\mu_2\ne0$ implies $C_2\ne0$.
Therefore since $\tilde L=0$ according to Lemma \ref{lem:ISP-mf=2}
we obtain the following two configurations
\begin{gather*}
\begin{aligned}
 {\scriptstyle\copyright,\copyright};\,
\widehat{\binom{1}{2}}\,{\overset{
{}_\curvearrowright}P_{\curlywedge}}E {\overset{
{}_\curvearrowleft}P_{\curlywedge}}-
H,\,\overline{\binom{1}{1}} SN : \text{ Example }
\Rightarrow &(a= 1,\; c=0,\;  g= 0,\; h=1) \\
& (\text{if }\widetilde M\ne0);
\end{aligned}
\\
 {\scriptstyle\copyright,\copyright};\,
\widehat{\binom{2}{3}}\,  {\overset{
{}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P- {\overset{
{}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P  : \text{ Example } \Rightarrow
(a=1,\; c=0,\; g= 0,\; h=0) \quad (\text{if }\widetilde M=0).
\end{gather*}
As all   possible cases are examined, we have proved that the
family of systems with two complex distinct finite singularities
possesses exactly  16 geometrically distinct global configurations
of singularities.


\subsection{The family of quadratic differential systems with  two
real  distinct finite singularities which in additional are
elemental}
 Assume that quadratic systems \eqref{sys:QSgen}   possess two real finite
singular points and both are  elemental, i.e. by \cite{Vlp-NATMA}
the conditions $\mu_0=\mu_1=0,$ $\mu_2\ne0$ and $\mathbf{U}>0$ hold.
According to \cite{Vlp-NATMA}  (see Table 2)  we have to consider
two cases: $\widetilde K\ne0$ and $\widetilde K=0$.


\subsubsection{Systems with $\widetilde K\ne0$.} In this case
according to \cite{Vlp-NATMA} we   consider the   family of
systems
\begin{equation} \label{sys:14a}
\dot x= cx+dy-cx^2+ 2 d u x y,\quad \dot y  = ex+fy-ex^2+2fuxy,
\end{equation}
which possess  the singular points $M_1(0,0)$ and $M_2(1,0)$.


\begin{remark} \label{rem:CS-(0,0)} \rm
Assume that we have a family  of quadratic systems
possessing a real elemental singular point for all values of the
parameters. Then by a translation of axes we may suppose this
point to be placed at the origin. In case we have one other real
elemental singularity (or even two such singularities), then we
can always use a linear transformation to place the second
singularity or even two such singularities) in specific positions
(for example at $(1,0)$ and in case a second such singular point
exists to place it at  $(0,1)$ ).  We arrive thus at a certain
normal form for the family, dictated by the position of these
singularities. Suppose that in the course of the study of this
family, under certain conditions on parameters expressed in
invariant form i.e. in terms of invariant polynomials, we find
that an elemental real singularity of the systems has a certain
geometric property, for example it is a node. Then we may always
suppose this singularity to be placed at the origin. This is clear
if we have just one real elemental singularity. If we have other
real elemental singular points then by the argument above we can
exchange its position with one of the other elemental singular
points via an affine transformation without changing the aspect of
the normal form.
\end{remark}

For   systems \eqref{sys:14a} calculations yield
\begin{equation} \label{val:14a}
\begin{gathered}
\mu_0=\mu_1=0,\quad \mu_2= (cf-d e)^2 (1 + 2 u) x^2,\\
 \widetilde K=4 (d e - c f) u x^2,\quad   \kappa= 128 d^2 (cf-d e) u^3.
\end{gathered}
\end{equation}
We remark that for the above systems the condition
$\mu_2 \widetilde{K}\ne0$ holds.  So in what follows we assume that the following
condition is satisfied
\begin{equation} \label{cond:14a}
(cf-d e) (1 + 2 u)u\ne0.
\end{equation}

\begin{remark} \label{rem:14a-d=1,0} \rm
We observe that the family of systems
\eqref{sys:14a} depends on five parameters. However due to a
rescaling we can reduce the number of the parameters to three. More
precisely since according to condition \eqref{cond:14a} we have
$d^2+f^2\ne0$, then  we may assume $(d,f)\in\{(1,1), (1,0), (0,1)\}$
due to the rescaling:
\begin{itemize}
\item[(i)] $(x,y,t)\mapsto(x,fy/d,t/f)$ if
$df\ne0$;
\item[(ii)] $(x,y,t)\mapsto(x, y/d,t)$ if $f=0$, and
\item[(iii)]
$(x,y,t)\mapsto(x, y,t/f)$ if $d=0$.
\end{itemize}
 \end{remark}

Considering \eqref{val:14a} and \eqref{cond:14a}   the condition $d\ne0$ is
equivalent to $\kappa\ne0$.


\subsubsection*{The case $\kappa\ne0$.} Then considering  Remark
\ref{rem:14a-d=1,0} we examine   the subfamilies of systems
\eqref{sys:14a} with  $(d,f)=(1,1)$  and  $(d,f)=(1,0)$.

\subsubsection*{A. Systems with $d=f=1$.}
We consider the 3-parameter family of systems:
\begin{equation} \label{sys:14a)-d=f=1}
\begin{gathered}
\dot x= cx+ y-cx^2+ 2   u x y,\quad (c - e) (1 + 2 u)u\ne0,\\
 \dot y = ex+ y-ex^2+2 uxy,
\end{gathered}
\end{equation}
for which calculations yield
\begin{equation} \label{val:14a)-d=f=1}
\begin{gathered}
\mu_0=\mu_1=0,\quad \mu_2= (c - e)^2 (1 + 2 u) x^2 ,\quad
 \widetilde K=4  (e  - c) u x^2,\\
\kappa= 128(c - e) u^3, \quad
 \eta=4  u^2 \big[(c - 2 u)^2 + 8 (c - e) u\big],\\
 F_2=(c - e)^2 (1 + 2 u) x^2,\quad G_8= 2(c - e)^2 u^2 (1 + 2 u),\\
  W_4=16 (c - e)^4 u^4 (1 + 2 u)^2 \big[(c-1)^2
+ 4 e\big] \big[(1 - c + 2 u)^2 + 4 (c - e) (1 + 2 u)\big],\\
  W_3=32 (c - e)^4 u^4 (1 + 2 u)^2 (1 + c^2 + 2 u + 2 c u - 4 e u + 2
  u^2),\\
   \widetilde M=-8 \big[( c + 2 u)^2 - 6 e u\big] x^2 + 16 u (c + 2 u) x y - 32 u^2
  y^2,\\
   \mathcal{T}_4= 4 (1 + c) (c - e)^2 ( 1 - c + 2 u) u^2 (1 + 2   u).
\end{gathered}
\end{equation}


\begin{remark} \label{rem:kappa-ne0->M-ne0}\rm
We observe that the condition $\mu_2\kappa\ne0$ gives $ \widetilde
MF_2G_8\ne0$ and the condition $\kappa>0$ implies $\eta>0$.
Moreover we have
$$
\operatorname{sign}(\widetilde K)=-\operatorname{sign}(\kappa),\quad \operatorname{sign}(G_8)=\operatorname{sign}(
\mu_2)=\operatorname{sign}(F_2).
$$
\end{remark}

\subsubsection*{The subcase $\kappa<0$.} Then by Remark
\ref{rem:kappa-ne0->M-ne0} we obtain $\widetilde K>0$.
\smallskip

(1) {\it The possibility $\mu_2<0$.} Then $1+2u<0$, i.e.
$u<-1/2$ and considering \eqref{val:14a)-d=f=1} we obtain $G_8<0$.
So since $\widetilde K>0$, according to \cite{Art-Llib-Vlp-IJBCh} (see
Table 1, lines 165-170) both finite singularities are
anti-saddles.
\smallskip

(a) {\it Assume first $W_4<0$.} Then we have a node and a focus and
whether the focus is a weak one or  not depends on the invariant
polynomial $\mathcal{T}_4$.   On the other hand due to $W_4\ne0$ we
have a generic node.
\smallskip

(a.1)  {\it The case $\mathcal{T}_4\ne0$.} Then by
\cite{Vlp-NATMA} the focus is strong.
\smallskip
($\alpha)$ {\it The subcase $\eta<0$.}  Then at infinity
we have one real and two complex singularities and as
 $\mu_2<0$ and $\kappa\ne0$  considering Lemma
\ref{lem:ISP-mf=2} we get the  global configuration of
singularities
\[
 n, f;\, \overline{\binom{2}{1}}S,
\copyright,\copyright :  \text{ Example } \Rightarrow (c=5,\;e= -1,\; u=-2).
\]

($\beta)$ {\it The subcase $\eta>0$.}  In this case at
infinity we have three real  singularities. As $\kappa<0$ and
$\mu_2<0$, by  Lemma \ref{lem:ISP-mf=2} we get the configuration
\[
 n, f;\, \overline{\binom{2}{1}}S, S, N^\infty:
\text{ Example } \Rightarrow  (c=1/2,\; e= -1/5,\; u=-2).
\]

($\gamma$) {\it The subcase $\eta=0$.}  In this case
considering Remark \ref{rem:kappa-ne0->M-ne0} we have  $\widetilde
M\ne0$. As   $\kappa\ne0$ and $\mu_2<0$, considering
 Lemma \ref{lem:ISP-mf=2}  we get the  global configuration
of singularities
\[
 n, f;\,\overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}} S :  \text{ Example }
\Rightarrow (c=3,\; e= -1/16,\; u=-2).
\]

(a.2)  {\it  The case $\mathcal{T}_4=0$.} Then by
\cite{Vlp-NATMA} the focus is  weak. Considering
\eqref{val:14a)-d=f=1} and the condition \eqref{cond:14a}, the
condition $\mathcal{T}_4=0$ gives $(c+1)(2u+1-c)=\rho_1\rho_2=0$.
By Remark \ref{rem:CS-(0,0)}    we may assume without loss of
generality that $\rho_1=0$, i.e. $c=-1$. Then for systems
\eqref{sys:14a)-d=f=1} we calculate:
\begin{equation} \label{val:T3,F1,W4->c=-1}
\begin{gathered}
\mathcal{T}_3=8 (1 + e)^2 u^2 (1 + u) (1 + 2 u),\quad
\mathcal{F}_1=2 (1 + e) (u-1) (1 + 2 u),\\
\mu_2=(1 + e)^2 (1 + 2 u) x^2,\quad \kappa=-128 (1 + e) u^3,\\
W_4=-256 (1 + e)^5 u^4 (1 + 2 u)^2 (e + 2 e u - u^2).
\end{gathered}
\end{equation}
We observe that the condition $\mu_2<0$  implies
$\mathcal{F}_1\ne0$. Moreover as $W_4<0$  we have
$\mathcal{T}_3\ne0$, otherwise we get $u=-1$  and this gives
$W_4=256(1 + e)^6\ge0$. Therefore by \cite{Vlp-NATMA}  the weak
focus has order one, and according  to
 Lemma \ref{lem:ISP-mf=2}  we obtain the following three global configurations
of singularities:
\begin{gather*}
 n, f^{(1)};\, \overline{\binom{2}{1}} S,
\copyright,\copyright :  \text{ Example } \Rightarrow
(c=-1,\; e= -3/2,\; u=-6/10) \quad (\text{if }\eta<0);
\\
\begin{aligned}
 n, f^{(1)};\, \overline{\binom{2}{1}} S, S,
N^\infty:  \text{ Example } \Rightarrow &(c=-1,\; e=  -51/50,\;
u=-7/10)  \\
& (\text{if }\eta>0);
\end{aligned}\\
\begin{aligned}
n, f^{(1)};\, \overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}} S:  \text{ Example }
\Rightarrow &(c=-1,\; e=   -36/35,\; u=-7/10) \\
& (\text{if } \eta=0).
\end{aligned}
\end{gather*}

(b) {\it  Suppose now $W_4>0$.} In this case as
$\widetilde K>0$ and $G_8<0$, according to
\cite{Art-Llib-Vlp-IJBCh} systems
\eqref{sys:14a)-d=f=1} possess two nodes if $W_3>0$ and two foci
or/and centers if $W_3<0$.
\smallskip

(b.1)  {\it  The case $W_3<0$.}
\smallskip

($\alpha$) {\it The subcase $\mathcal{T}_4\ne0$.} Then by
\cite{Vlp-NATMA} both   foci are   strong. Thus considering the
conditions $\mu_0=\mu_1=0$, $\mu_2<0$, $\kappa\ne0$  and Lemma
\ref{lem:ISP-mf=2}  we arrive at the following three global
configurations of singularities:
\begin{gather*}
 f, f;\, \overline{\binom{2}{1}} S,
\copyright,\copyright :  \text{ Example } \Rightarrow
 (c=1,\; e= -1,\; u=-2) \quad (\text{if }\eta<0);
\\
 f, f;\, \overline{\binom{2}{1}} S, S, N^\infty:
\text{ Example } \Rightarrow \quad (c=1 ,\; e= -1/2,\; u=-2)\quad
(\text{if } \eta>0);
\\
f, f;\,\overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}} S :  \text{ Example }
\Rightarrow (c=1,\; e= - 9/16,\; u=-2)\quad (\text{if }\eta=0).
\end{gather*}

($\beta$) {\it The subcase $\mathcal{T}_4=0$.} As it was
mentioned earlier we may assume $c=-1$. Then at least one focus is a
weak one.
\smallskip

($\beta.1$) {\it The possibility $\mathcal{T}_3\ne0$.}
In this case only one focus is weak. Moreover considering
\eqref{val:T3,F1,W4->c=-1} we observe that the condition $\mu_2<0$
implies $\mathcal{F}_1\ne0$ and the weak focus could only be of
the first order. So in view of the arguments above and Lemma
\ref{lem:ISP-mf=2} we get three global configurations of
singularities:
\begin{gather*}
f, f^{(1)};\, \overline{\binom{2}{1}} S,
\copyright,\copyright :  \text{ Example } \Rightarrow
(c=-1,\; e=-2,\; u=-3/2)\quad (\text{if }\eta<0);
\\
 f, f^{(1)};\, \overline{\binom{2}{1}} S, S,
N^\infty:  \text{ Example } \Rightarrow
 (c=-1 ,\; e=-5/4 ,\; u=-3/2)\quad (\text{if }\eta>0);
\\
 f, f^{(1)};\,\overline{\binom{0}{2}}SN,\overline{\binom{2}{1}} S :  \text{ Example }
\Rightarrow (c=-1,\; e=-4/3 ,\; u=- 3/2)\quad (\text{if }\eta=0).
\end{gather*}

($\beta.2$) {\it The possibility $\mathcal{T}_3=0$.}
Considering \eqref{val:T3,F1,W4->c=-1} we get $u=-1$ and then we
calculate:
\begin{gather*}
\mathcal{T}_4=\mathcal{T}_3=\mathcal{F}=0,\quad
\mathcal{T}_2=-4 (1 + e)^2,\quad \mathcal{F}_1=4 (1 + e),\\
\mu_2=-(1 + e)^2 x^2,\quad \kappa=128 (1 + e),\quad \eta=4 (9 + 8 e).
\end{gather*}
So the condition $\mu_2\ne0$ gives $\mathcal{F}_1\ne0$ and
according to \cite{Vlp-NATMA} we have two first order weak singularities,
which in this case are foci.

Thus considering  Lemma \ref{lem:ISP-mf=2}  we obtain the
following three  global configurations of singularities
\begin{gather*}
f^{(1)}, f^{(1)};\, \overline{\binom{2}{1}} S,
\copyright,\copyright :  \text{ Example } \Rightarrow
(c=-1,\; e= -2,\; u=-1) \quad (\text{if } \eta<0);
\\
 f^{(1)}, f^{(1)};\, \overline{\binom{2}{1}} S,
S, N^\infty:  \text{ Example } \Rightarrow
 (c=-1,\; e=-17/16,\; u=-1) \quad (\text{if }\eta>0);
\\
 f^{(1)}, f^{(1)};\,\overline{\binom{0}{2}}SN,\overline{\binom{2}{1}} S :  \text{ Example }
\Rightarrow (c=-1,\; e= -9/8,\; u=-1)\quad (\text{if } \eta=0).
\end{gather*}

(b.2)  {\it  The case $W_3>0$.} According to
\cite{Art-Llib-Vlp-IJBCh} (see Table 1, line 165) we have two
nodes and both are generic (due to $W_4\ne0$). According to
 Lemma \ref{lem:ISP-mf=2}   we arrive at  the following three  global
configurations of singularities:
\begin{gather*}
n, n;\, \overline{\binom{2}{1}} S,
\copyright,\copyright :  \text{ Example } \Rightarrow
 (c=6,\; e= -1/3,\; u=-2) \quad (\text{if }\eta<0);
\\
n, n ;\, \overline{\binom{2}{1}} S, S,
N^\infty:  \text{ Example } \Rightarrow
(c=6,\; e=  -1/5,\; u=-2) \quad (\text{if }\eta>0);
\\
n, n ;\, \overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}} S:  \text{ Example }
\Rightarrow (c=6,\; e= -1/4,\; u=-2) \quad (\text{if }\eta=0).
\end{gather*}

(c) {\it   Admit finally  $W_4=0$.} Then we have a node
with coinciding eigenvalues and  by Remark \ref{rem:CS-(0,0)}
without loss of generality we may assume that $M_1(0,0)$ is such a
point. So the corresponding discriminant $\tau_1=(c-1)^2 + 4 e=0$, and we obtain $e=-(c-1)^2/4$, and in this case  calculations yield:
\begin{equation} \label{val:W4=0}
\begin{gathered}
 \mu_2= (1 + c)^4 (1 + 2 u) x^2/16 ,\quad
 \kappa= 32 (1 + c)^2 u^3,\quad
 \eta=4 u^2 (1 + 2 u) (c^2 + 2 u),\\
  W_4=0,\quad
  W_3=(1 + c)^8 u^4 (1 + u) (1 + 2 u)^2 (1 + c^2 + 2 u)/8,\\
    \mathcal{T}_4=-(1 + c)^5 (c-1   - 2 u) u^2 (1 + 2 u)/4.
\end{gathered}
\end{equation}

(c.1) {\it The case $W_3<0$.} According to
\cite{Art-Llib-Vlp-IJBCh} (see Table 1, line 168) we have one
focus and  one node which  is $n^d$ (as the Jacobian matrix is not
diagonal).
\smallskip

($\alpha$) {\it The subcase $\mathcal{T}_4\ne0$.}  Then
the focus is strong and we get the following three  global
configurations of singularities
\begin{gather*}
n^d, f;\, \overline{\binom{2}{1}} S,
\copyright,\copyright :  \text{ Example } \Rightarrow
 (c=2,\; e= -1/4,\; u=-3/2) \quad (\text{if }\eta<0);
\\
 n^d, f;\, \overline{\binom{2}{1}} S, S,
N^\infty:  \text{ Example } \Rightarrow
 (c=2,\; e= -1/4,\; u=-9/4) \quad (\text{if }\eta>0);
\\
n^d, f;\,\overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}} S :  \text{ Example }
\Rightarrow (c=2,\; e= -1/4,\; u=-2)\quad (\text{if }\eta=0).
\end{gather*}

($\beta$) {\it The subcase $\mathcal{T}_4=0$.} Considering
\eqref{val:W4=0} and the condition $\mu_2\kappa\ne0$ we obtain $(c-1
- 2 u)=0$, i.e. $c=2u+1$. Then we calculate
\begin{equation} \label{val:W4=T4=0}
\begin{gathered}
\mathcal{T}_3=8 u^2 (1 + u)^5 (1 + 2 u),\quad
 \mathcal{F}_1=-2 (1 + u)^2 (1 + 2 u) (1 + 3 u),\\
\eta=4 u^2 (1 + 2 u) (1 + 6 u + 4 u^2),\quad
 \mu_2 =(1 + u)^4 (1 + 2 u) x^2,\\
\kappa=128 u^3 (1 + u)^2,\quad W_4=0,\quad
 W_3=64 u^4 (1 + u)^{10} (1 + 2 u)^3.
\end{gathered}
\end{equation}
So the conditions $W_3\ne0$ and $\mu_2<0$ implies
$\mathcal{T}_3\mathcal{F}_1\ne0$ and according to \cite{Vlp-NATMA}
the focus $M_2(1,0)$ is a weak focus of the first order. Therefore
considering  Lemma \ref{lem:ISP-mf=2}  we arrive at the following
three global configurations of singularities
\begin{gather*}
 n^d, f^{(1)};\, \overline{\binom{2}{1}} S,
\copyright,\copyright :  \text{ Example } \Rightarrow
 (c=-2,\; e= -9/4,\; u=-3/2) \quad (\text{if }\eta<0);
\\
\begin{aligned}
 n^d, f^{(1)};\, \overline{\binom{2}{1}} S, S,
N^\infty:  \text{ Example } \Rightarrow
&(c=-7/5 ,\; e= -36/25,\; u=-6/5) \\
& (\text{if }\eta>0);
\end{aligned}\\
\begin{aligned}
 n^d, f^{(1)};\,\overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}} S :  \text{ Example }
\Rightarrow  &(c=2 u + 1,\; e= -  (c-1)^2/4,\\
& u=-(3+\sqrt{5})/4)\quad (\text{if }\eta=0).
\end{aligned}
\end{gather*}

(c.2)  {\it The case $W_3>0$.} According to
\cite{Art-Llib-Vlp-IJBCh} (see Table 1, line 166) we have two
nodes (one of them being $n^d$).

Therefore considering  Lemma \ref{lem:ISP-mf=2}   we get the
following three  global configurations of singularities:
\begin{gather*}
 n, n^d;\, \overline{\binom{2}{1}} S,
\copyright,\copyright :  \text{ Example } \Rightarrow
(c=4/3,\; e= -1/36,\; u=-8/10) \quad (\text{if }\eta<0);
\\
 n, n^d;\, \overline{\binom{2}{1}} S, S,
N^\infty:  \text{ Example } \Rightarrow
(c=4/3,\; e= -1/36,\; u=-17/18) \quad (\text{if }\eta>0);
\\
 n, n^d;\,\overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}} S :  \text{ Example }
\Rightarrow (c=4/3,\; e= -1/36,\; u=-8/9)\quad (\text{if }\eta=0).
\end{gather*}


(c.3)   {\it The case $W_3=0$.} Considering \eqref{val:W4=0} and
the condition $\mu_2<0$ we obtain $(1 + u)(1 + c^2 + 2 u)=0$ and
in this case we have two nodes $n^d$ (as no one of the Jacobian
matrices is  diagonal).

\smallskip
($\alpha$) {\it The subcase $u=-1$.} We have
$$
\eta= 4(2-c^2),\quad \mu_2=-(1 + c)^4 x^2/16<0,\quad
 \kappa=-32 (1 + c)^2<0.
$$
So  considering  Lemma \ref{lem:ISP-mf=2}  we obtain the following
three  global configurations of singularities:
\begin{gather*}
 n^d, n^d;\, \overline{\binom{2}{1}} S,
\copyright,\copyright :  \text{ Example } \Rightarrow
 (c=2,\; e= -1/4,\; u=-1) \quad (\text{if }\eta<0);
\\
 n^d, n^d;\, \overline{\binom{2}{1}} S, S,
N^\infty:  \text{ Example } \Rightarrow
 (c=1/2 ,\; e= -1/16,\; u=-1) \quad (\text{if }\eta>0);
\\
\begin{aligned}
 n^d, n^d;\,\overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}} S :  \text{ Example }
\Rightarrow &(c=\sqrt{2},\; e= -  (c-1)^2/4,\; u=-1)\\
&(\text{if }\eta=0).
\end{aligned}
\end{gather*}


($\beta$) {\it The subcase $u=-(c^2+1)/2$.} We calculate
$$
\eta= c^2 (1 + c^2)^2,\quad \mu_2=-c^2 (1 + c)^4 x^2/16<0,\quad
\kappa=-4 (1 + c)^2 (1 + c^2)^3<0
$$
and in this case we could only have $\eta>0$ and such a
configuration was detected above.
\smallskip

(2) {\it The possibility $\mu_2>0$.} Then  by
\eqref{val:14a)-d=f=1} we have $1+2u>0$ and this implies $G_8>0$.
Since $\widetilde K>0$ according to \cite{Art-Llib-Vlp-IJBCh}
systems \eqref{sys:14a)-d=f=1} possess a saddle and a focus (or a center)
if $W_4<0$ and a saddle and a node if $W_4\ge0$.
\smallskip

(a) {\it Assume first  $W_4<0$,} i.e. we have a saddle
and a focus.
\smallskip

(a.1) {\it The subcase $\mathcal{T}_4\ne0$.} Then by
\cite{Vlp-NATMA} we could not have weak singularities. So
considering  Remark \ref{rem:kappa-ne0->M-ne0} and
 Lemma \ref{lem:ISP-mf=2}  we get the following three
global configurations of singularities
\begin{gather*}
 s, f;\, \overline{\binom{2}{1}} N,
\copyright,\copyright :  \text{ Example } \Rightarrow
(c=-8,\; e= 2,\; u=2) \quad (\text{if }\eta<0);
\\
 s, f;\, \overline{\binom{2}{1}} N, S, N^\infty:
 \text{ Example } \Rightarrow (c=-8 ,\; e= 1/2,\; u=2) \quad
(\text{if }\eta>0);
\\
s, f;\,\overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}} N :  \text{ Example }
\Rightarrow (c=-8,\; e= 1,\; u=2) \quad (\text{if }\eta=0).
\end{gather*}

(a.2) {\it The subcase $\mathcal{T}_4=0$.} As it was
shown earlier we can assume $c=-1$ and  we calculate
$$
\mathcal{T}_3\mathcal{F}=8 (1 + e)^3 u^4 (1 + u)^2 (1 + 2 u)^2.
$$
Therefore  considering
\eqref{val:T3,F1,W4->c=-1} we conclude that the condition
$\kappa<0$ and $\mu_2>0$ imply $\mathcal{T}_3\mathcal{F}\ne0$.
Moreover, as $(1 + e) u>0$ (due to $\kappa<0$) from
\eqref{val:T3,F1,W4->c=-1} we have
\begin{equation} \label{val:sign(T3F)}
\operatorname{sign}(\mathcal{T}_3\mathcal{F})=\operatorname{sign}(u),\quad
\operatorname{sign}(W_4)=\operatorname{sign}(u(e + 2 e u - u^2)).
\end{equation}
\smallskip

($\alpha$) {\it The possibility
$\mathcal{T}_3\mathcal{F}<0$.} Then $u<0$ and according to
\cite{Vlp-NATMA} systems \eqref{sys:14a)-d=f=1} with $c=-1$ possess
a weak focus. As $-1/2<u<0$ then $\mathcal{F}_1\ne0$ and we have a
weak first order focus. So considering  Lemma \ref{lem:ISP-mf=2} we
get the following three  global configurations of singularities
\begin{gather*}
s, f^{(1)};\, \overline{\binom{2}{1}} N,
\copyright,\copyright :  \text{ Example } \Rightarrow
(c=-1,\; e= -2,\; u=-1/4) \quad (\text{if }\eta<0);
\\
 s, f^{(1)};\, \overline{\binom{2}{1}} N, S,
N^\infty: \text{ Example } \Rightarrow (c=-1 ,\; e= -17/16,\;
u=-1/4) \quad (\text{if }\eta>0);
\\
 s, f^{(1)};\,\overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}} N :  \text{ Example }
\Rightarrow ( c=-1,\; e=-9/8,\; u=-1/4)\quad (\text{if }\eta=0).
\end{gather*}

($\beta$) {\it The possibility $\mathcal{T}_3\mathcal{F}>0$.}
 In this case $u>0$ and according to
\cite{Vlp-NATMA} systems \eqref{sys:14a)-d=f=1} with $c=-1$ possess
a weak saddle.
\smallskip

($\beta.1$) {\it Assume first  $\mathcal{F}_1\ne0$.} The
weak saddle has order one and considering
 Lemma \ref{lem:ISP-mf=2}  we obtain the following three
 global configurations of singularities:
\begin{gather*}
 s^{(1)}, f;\, \overline{\binom{2}{1}} N,
\copyright,\copyright :  \text{ Example } \Rightarrow
(c=-1,\; e= 1,\; u=1/5) \quad (\text{if }\eta<0);
\\
s^{(1)}, f ;\, \overline{\binom{2}{1}} N, S,
N^\infty:  \text{ Example } \Rightarrow (c=-1 ,\; e= 1/5,\;
u=1/5) \quad (\text{if }\eta>0);
\\
s^{(1)}, f ;\,\overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}} N :  \text{ Example }
\Rightarrow ( c=-1,\; e= 9/40,\; u=1/5) \quad (\text{if }\eta=0).
\end{gather*}

($\beta.2$) {\it Suppose now  that $\mathcal{F}_1=0$.}
Then $u=1$ and we calculate
\begin{equation} \label{val:14a)-d=f=1=u}
\begin{gathered}
\mathcal{T}_3\mathcal{F}=288 (1 + e)^3,\quad \mathcal{F}_1=0, \quad
\mathcal{F}_2= -432 (1 + e)^2,   \\
\kappa=-128 (1 + e), \quad \eta= 4 ( 1 - 8 e),\quad
 W_4= 2304 (1 + e)^5 ( 1 - 3 e),
\end{gathered}
\end{equation}
and as $\kappa<0$ we get $e+1>0$ and hence  $\mathcal{F}_2\ne0$ .
Therefore as $\mathcal{T}_3\mathcal{F}>0$ and  $\mathcal{F}_1=0$
by \cite{Vlp-NATMA} we obtain a  weak saddle of order two.

On the other hand the condition $W_4<0$ due to $e+1>0$ gives
$e>1/3$ and this implies $\eta<0$.

Thus considering  Lemma \ref{lem:ISP-mf=2}  we obtain the
configuration
\[
 s^{(2)}, f;\, \overline{\binom{2}{1}} N,
\copyright,\copyright :  \text{ Example } \Rightarrow
(c=-1,\; e= 1,\; u=1).
\]

(b) {\it Suppose now that  $W_4>0$,} i.e. by
\cite{Art-Llib-Vlp-IJBCh} we have a saddle and a node.
\smallskip

(b.1) {\it The subcase $\mathcal{T}_4\ne0$.} Then by
\cite{Vlp-NATMA} we could not have weak singularities, i.e. the
saddle is strong.
So considering  Lemma \ref{lem:ISP-mf=2}  we obtain the following
three  global configurations of singularities:
\begin{gather*}
s, n;\, \overline{\binom{2}{1}} N,
\copyright,\copyright :  \text{ Example } \Rightarrow
(c=-2,\; e=2/5,\; u=2) \quad (\text{if }\eta<0);
\\
 s, n ;\, \overline{\binom{2}{1}} N, S,
N^\infty:  \text{ Example } \Rightarrow (c=-2 ,\; e= 1/5,\; u=2)
\quad (\text{if }\eta>0);
\\
s, n ;\,\overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}} N :  \text{ Example }
\Rightarrow ( c=-2,\; e=1/4,\; u=2)\quad (\text{if }\eta=0).
\end{gather*}

(b.2) {\it The subcase $\mathcal{T}_4=0$.} As it was shown
earlier we can assume $c=-1$ and considering
\eqref{val:T3,F1,W4->c=-1} we conclude that the conditions
$\kappa<0$ and $\mu_2>0$ imply $\mathcal{T}_3\ne0$.
\smallskip

($\alpha$) {\it Assume first  $\mathcal{F}_1\ne0$.} Then
the weak saddle has order one and considering
 Lemma \ref{lem:ISP-mf=2}  we obtain the following three
 global configurations of singularities:
\begin{gather*}
 s^{(1)}, n;\, \overline{\binom{2}{1}} N,
\copyright,\copyright :  \text{ Example } \Rightarrow
(c=-1,\; e=5/8,\; u=2) \quad (\text{if }\eta<0);
\\
 s^{(1)}, n ;\, \overline{\binom{2}{1}} N, S,
N^\infty:  \text{ Example } \Rightarrow (c=-1 ,\; e= 1/2,\;
u=2) \quad (\text{if }\eta>0);
\\
s^{(1)}, n ;\,\overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}} N :  \text{ Example }
\Rightarrow ( c=-1,\; e=9/16,\; u=2) \quad (\text{if }\eta=0).
\end{gather*}

($\beta$) {\it Suppose now  $\mathcal{F}_1=0$.}  In this
case we get $u=1$ and considering \eqref{val:14a)-d=f=1=u} we
deduce that the condition $\kappa<0$ implies $\mathcal{F}_2\ne0$.
Therefore according to \cite{Vlp-NATMA} we have a second order
weak saddle and we arrive at the following three  global
configurations of singularities:
\begin{gather*}
 s^{(2)}, n;\, \overline{\binom{2}{1}} N,
\copyright,\copyright :  \text{ Example } \Rightarrow
(c=-1,\; e=1/4,\; u=1) \quad (\text{if }\eta<0);
\\
s^{(2)}, n ;\, \overline{\binom{2}{1}} N, S,
N^\infty:  \text{ Example } \Rightarrow (c=-1 ,\; e= 1/9,\;
u=1) \quad (\text{if }\eta>0);
\\
s^{(2)}, n ;\,\overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}} N :  \text{ Example }
\Rightarrow  ( c=-1,\; e=1/8,\; u=1)\quad (\text{if }\eta=0).
\end{gather*}

(c) {\it Admit finally  $W_4=0$.} As it was mentioned above in this
case we may assume   $e=-(c-1)^2/4$. On the other hand as $G_8>0$
and $\widetilde K>0$, according to \cite{Art-Llib-Vlp-IJBCh} systems
\eqref{sys:14a)-d=f=1} possess a saddle and a node, which is $n^d$
(due to  $W_4=0$ and the fact that the jacobian is not diagonal).
According to \eqref{val:W4=0} the conditions $\kappa<0$ and
$\mu_2>0$ yield $-1/2<u<0$.
\smallskip

(c.1) {\it The subcase $\mathcal{T}_4\ne0$.} Then by
\cite{Vlp-NATMA} we could not have weak singularities, i.e. the
saddle is strong.
So considering  Lemma \ref{lem:ISP-mf=2}  we obtain the following
three  global configurations of singularities:
\begin{gather*}
s, n^d;\, \overline{\binom{2}{1}} N,
\copyright,\copyright :  \text{ Example } \Rightarrow
(c=1/2,\; e=-1/16,\; u=-1/5) \quad (\text{if }\eta<0);
\\
s, n^d ;\, \overline{\binom{2}{1}} N, S,
N^\infty:  \text{ Example } \Rightarrow (c=1/2 ,\; e=-1/16,\;
u=-1/9) \quad (\text{if }\eta>0);
\\
 s, n^d ;\,\overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}} N :  \text{ Example }
\Rightarrow ( c=1/2,\; e=-1/16,\; u=-1/8)\quad (\text{if }\eta=0).
\end{gather*}


(c.2) {\it The subcase $\mathcal{T}_4=0$.} Considering
\eqref{val:W4=0} and the condition $\mu_2\kappa\ne0$ we obtain
$c-1 - 2 u=0$, i.e. $c=2u+1$. Then we obtain
\eqref{val:W4=T4=0} and we can observe that $\mathcal{T}_3\ne0$
(due to $\mu_2\kappa\ne0$).
\smallskip

($\alpha$) {\it The possibility  $\mathcal{F}_1\ne0$.}
Then by \cite{Vlp-NATMA} the weak saddle has order one and
considering  Lemma \ref{lem:ISP-mf=2}  we obtain the following
three global configurations of singularities:
\begin{gather*}
s^{(1)}, n^d;\, \overline{\binom{2}{1}} N,
\copyright,\copyright :  \text{ Example } \Rightarrow
(c=1/2,\; e=-1/16,\; u=-1/4) \quad (\text{if }\eta<0);
\\
 s^{(1)}, n^d ;\, \overline{\binom{2}{1}} N, S,
N^\infty:  \text{ Example } \Rightarrow (c=2/3 ,\; e=-1/36,\;
u=-1/6) \,\, (\text{if }\eta>0);
\\
\begin{aligned}
s^{(1)}, n^d ;\,\overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}} N :  \text{ Example }
\Rightarrow &(c=2u+1,\; e=-(c-1)^2 /4,\\
& u=(\sqrt{5}-3)/4) \quad (\text{if }\eta=0).
\end{aligned}
\end{gather*}

($\beta$) {\it The possibility  $\mathcal{F}_1=0$.}
Considering \eqref{val:W4=T4=0} we obtain $u=-1/3$ and then we have
$$
\mathcal{T}_4=\mathcal{F}_1=0, \quad \mathcal{T}_3 =256/6561, \quad
\mathcal{F}_2=-256/19683, \quad \eta=-20/243.
$$
Hence by \cite{Vlp-NATMA} the saddle is of the second order. As
$\eta<0$ considering  Lemma \ref{lem:ISP-mf=2}  we obtain the
 configuration
$$
s^{(2)}, n^d;\, \overline{\binom{2}{1}} N,
\copyright,\copyright :  \text{ Example } \Rightarrow
 (c=1/3,\; e=-1/9,\; u=-1/3).
$$


\subsubsection*{The subcase $\kappa>0$.} According to
\eqref{val:14a)-d=f=1} and Remark \ref{rem:kappa-ne0->M-ne0} the
condition $\kappa>0$  implies $\eta>0$ and $\widetilde K<0$, and we
consider two possibilities: $\mu_2<0$ and $\mu_2>0$.
\smallskip

(1) {\it The possibility $\mu_2<0$.} Then $u<-1/2$ and
considering \eqref{val:14a)-d=f=1} we obtain $F_2<0$ and $G_8<0$.
As $\widetilde K<0$   according to \cite{Art-Llib-Vlp-IJBCh} (see
Table 1, line  148) both finite singularities are saddles.

On the other hand according to  Lemma \ref{lem:ISP-mf=2} due to
$\eta>0$, $\kappa>0$ and $\mu_2<0$ at infinity we have the
configuration $\overline{\binom{2}{1}}N, N^f, N^f$.
\smallskip

(b) {\it Assume first  $\mathcal{T}_4\ne0$,} i.e. both saddles
are strong and this leads to the  global configuration  of
singularities
\[
s, s ;\, \overline{\binom{2}{1}} N, N^f, N^f:
\text{ Example } \Rightarrow (c=-2 ,\; e=0,\; u=-2).
\]

(b) {\it  Suppose now  $\mathcal{T}_4=0$.}
 As it was shown earlier we can assume $c=-1$ and we consider
 \eqref{val:T3,F1,W4->c=-1}.
\smallskip

(b.1)  {\it The  case $\mathcal{T}_3\ne0$.} In this situation only one saddle
is weak and as $\mathcal{F}_1\ne0$ (due to $\mu_2<0$), according to
\cite{Vlp-NATMA} the order of the weak saddle is one and we get
the   configuration
\[
s, s^{(1)} ;\, \overline{\binom{2}{1}} N, N^f,
N^f:  \text{ Example } \Rightarrow (c=-1 ,\; e=1,\; u=-2) .
\]

(b.2)  {\it The  case $\mathcal{T}_3=0$.} Then $u=-1$ and we have
\[
\mathcal{T}_4 =\mathcal{T}_3 =\mathcal{F}=0,\quad
\mathcal{T}_2 =-4 (1 + e)^2,\quad \mathcal{F}_1= 4 (1 + e),\quad
\mu_2=-(1 + e)^2 x^2\ne0.
\]
Hence $\mathcal{T}_2<0$ and as $\mathcal{F}_1\ne0$ according to
\cite{Vlp-NATMA} we have two weak saddles each one of the first
order
\[
s^{(1)}, s^{(1)} ;\, \overline{\binom{2}{1}} N,
N^f, N^f:  \text{ Example } \Rightarrow (c=-1 ,\; e=1,\; u=-1).
\]

(2) {\it The possibility $\mu_2>0$.} In this case we have
$u>-1/2$ and considering \eqref{val:14a)-d=f=1} we obtain $F_2>0$
and $G_8>0$. As $\widetilde K<0$   according to
\cite{Art-Llib-Vlp-IJBCh}  (see Table 1, lines  149,156,161) we
have a saddle and an anti-saddle. The type of the anti-saddle is
governed by invariant polynomial $W_4$.

On the other hand due to the condition $\eta>0$, $\kappa>0$ ,
$\mu_2>0$ and considering Remark \ref{rem:kappa-ne0->M-ne0},
according to  Lemma \ref{lem:ISP-mf=2}  at infinity we have the
configuration $\overline{\binom{2}{1}}S, N^f, N^f$.
\smallskip

(a) {\it Assume first  $W_4<0$,} i.e. we have a saddle and a
focus or a center.
\smallskip

(a.1) {\it The case $\mathcal{T}_4\ne0$.} Then by
\cite{Vlp-NATMA} we could not have weak singularities i.e. the
saddle and the focus are both strong ones. Thus  we get the global
configuration of singularities
\[
 s, f ;\, \overline{\binom{2}{1}} S, N^f, N^f:
\text{ Example } \Rightarrow (c= 1 ,\; e=-1,\; u=2).
\]

(a.2) {\it The subcase $\mathcal{T}_4=0$.} As it was shown
earlier we can assume $c=-1$ and considering
\eqref{val:T3,F1,W4->c=-1} we conclude that the condition
$\kappa>0$ and $\mu_2>0$ imply $\mathcal{T}_3\ne0$. Moreover as
$(1+e)u<0$ (due to $\kappa>0$)    we have
$\operatorname{sign}(\mathcal{T}_3\mathcal{F})=-\operatorname{sign}(u)$.
\smallskip

($\alpha$) {\it The possibility
$\mathcal{T}_3\mathcal{F}<0$.} Then $u>0$ and in this case according
to \cite{Vlp-NATMA} systems \eqref{sys:14a)-d=f=1} with $c=-1$
possess a weak focus.
\smallskip

($\alpha.1$)  {\it Assume first $\mathcal{F}_1\ne0$.} In
this case by \cite{Vlp-NATMA} the weak  focus is of order one and
this leads to the configuration
\[
s, f^{(1)} ;\, \overline{\binom{2}{1}} S, N^f, N^f:  \text{ Example }
\Rightarrow (c= -1 ,\; e=-2,\; u=2) .
\]

($\alpha.2$) {\it Admit now that  $\mathcal{F}_1=0$.} Then
$u=1$, $e<-1$ and this implies  $\mathcal{F}_2=-432 (1 + e)^2\ne0$.
So by \cite{Vlp-NATMA}  we could  have a weak focus of the order at
most two and this leads to the configuration
\[
s, f^{(2)} ;\, \overline{\binom{2}{1}} S, N^f,
N^f:  \text{ Example } \Rightarrow (c= -1 ,\; e=-2,\; u=1).
\]


($\beta$) {\it The possibility
$\mathcal{T}_3\mathcal{F}>0$.} Then $u<0$ and in this case according
to \cite{Vlp-NATMA} systems \eqref{sys:14a)-d=f=1} with $c=-1$
possess a weak saddle. As $-1/2<u<0$ then $\mathcal{F}_1\ne0$ and we
have a weak saddle of order one. Thus we get the
configuration
\[
s^{(1)}, f;\, \overline{\binom{2}{1}}S,
N^f, N^f:  \text{ Example } \Rightarrow (c= -1 ,\; e=1/2,\;
u=-1/4).
\]

(b) {\it Suppose now  $W_4>0$.} By \cite{Art-Llib-Vlp-IJBCh}
besides the saddle we have a node (which is generic due to
$W_4\ne0$).
\smallskip

(b.1) {\it The subcase $\mathcal{T}_4\ne0$.} Then by
\cite{Vlp-NATMA} we could not have weak singularities and this
leads to the  configuration  of singularities
\[
s, n ;\, \overline{\binom{2}{1}}S, N^f, N^f:  \text{ Example }
\Rightarrow (c= 2 ,\; e= 1,\; u=2).
\]

(b.2) {\it The subcase $\mathcal{T}_4=0$.} As it was shown
earlier we can assume $c=-1$ and considering
\eqref{val:T3,F1,W4->c=-1} we conclude that the condition
$\kappa>0$ and $\mu_2>0$ imply $\mathcal{T}_3\ne0$. Moreover we
claim that   in this case the condition $\mathcal{F}_1\ne0$ holds.
Indeed assuming $\mathcal{F}_1=0$ by \eqref{val:T3,F1,W4->c=-1} we
obtain  $u=1$ and then the conditions
$$
\kappa=-128 (1 + e)>0,\quad W_4= 2304 (1 + e)^5 ( 1 - 3 e)>0
$$
are incompatible. Thus $\mathcal{F}_1\ne0$ and we have a weak
saddle of   order one, i.e. we get the configuration
\[
s^{(1)}, n ;\, \overline{\binom{2}{1}} S, N^f,
N^f:  \text{ Example } \Rightarrow (c= -1 ,\; e= -1/2,\;
u=-1/3).
\]

(c) {\it Admit finally that  $W_4=0$.}  As it was mentioned above
in this case we may assume   $e=-(c-1)^2/4$ (i.e. $\tau_1=0$). On
the other hand as $G_8>0$,  $F_2>0$ and $\widetilde K<0$ according
to \cite{Art-Llib-Vlp-IJBCh} (see Table 1, line 151) systems
\eqref{sys:14a)-d=f=1} possess a saddle and a   node, which is
$n^d$ (due to  $W_4=0$ and the non-diagonal corresponding
matrix of the linearization). According to
\eqref{val:W4=0} the condition  $\kappa>0$   yields $ u>0$.
\smallskip

(c.1) {\it The subcase $\mathcal{T}_4\ne0$.} Then by
\cite{Vlp-NATMA} we could not have weak singularities, i.e. the
saddle is strong. So   we get the   configuration
\[
s, n^d ;\, \overline{\binom{2}{1}} S, N^f, N^f:
\text{ Example } \Rightarrow (c= 1 ,\; e=0,\; u=1).
\]

(c.2) {\it The subcase $\mathcal{T}_4=0$.} Considering
\eqref{val:W4=0} and the condition $\mu_2\kappa\ne0$ we get
$ c-1 - 2 u =0$, i.e. $c=2u+1$. Then we obtain \eqref{val:W4=T4=0}
and we  observe that $\mathcal{T}_3\ne0$ (due to
$\mu_2\kappa\ne0$). Moreover in this case we obtain
$ \mathcal{F}_1=-2 (1 + u)^2 (1 + 2 u) (1 + 3 u)\ne0 $ due to
$u>0$. Therefore we have a weak saddle of order one and this
leads to the configuration
\[
s^{(1)}, n^d ;\, \overline{\binom{2}{1}} S, N^f, N^f:  \text{ Example }
\Rightarrow (c= 3 ,\; e=-1,\; u=1).
\]

As all the cases have been examined the investigation of systems
\eqref{sys:14a)-d=f=1} is completed.

\subsubsection*{B. Systems with $d=1,\, f=0$.}
In this case for systems \eqref{sys:14a} with $d=1$ and $f=0$  we
calculate
$$
\kappa=-128 e u^3,\quad \mu_2=e^2 (1 + 2 u) x^2,\quad
 \mathcal{T}_4=-4 c^2 e^2 u^2 (1 + 2 u).
$$
We observe that due to $\mu_2\kappa\ne0$ the condition $c\ne0$ is
equivalent to $\mathcal{T}_4\ne0$.


\subsubsection*{The subcase $\mathcal{T}_4\ne0$.} Then $c\ne0$ and
we may assume $c=1$ due to  the rescaling $(x,y,t)\mapsto(x,cy,
t/c)$. So we get the 2-parameter family of systems
\begin{equation} \label{sys:14a)-d=1,f=0}
 \dot x=  x+y- x^2+ 2   u x y,\quad
 \dot y  = ex -ex^2,\quad eu(1+2u)\ne0,
\end{equation}
for which we calculate
\begin{equation} \label{val:14a)-d=1,f=0}
\begin{gathered}
\mu_0=\mu_1=0,\quad \mu_2= e^2 (1 + 2 u) x^2 ,\quad
 \widetilde K=4 e u x^2,\quad \kappa= -128 e u^3, \\
 \eta= 4 u^2 ( 1 - 8 e u),\quad
 F_2=e^2 (1 + 2 u) x^2,\quad G_8= 2 e^2 u^2 (1 + 2 u),\\
  W_4= 16 e^4 (1 + 4 e) u^4 (1 + 2 u)^2 ( 1 - 4 e - 8 e u),\\
  W_3= 32 e^4 u^4 (1 + 2 u)^2 ( 1 - 4 e u),\quad
   \widetilde M=8 (6 e u-1) x^2 + 16 u x y - 32 u^2 y^2.
\end{gathered}
\end{equation}

\begin{remark} \label{rem:kappa-ne0->M-ne0-f0} \rm
We observe that the condition $\mu_2\kappa\ne0$ gives
$ \widetilde MF_2G_8\ne0$. Moreover we have
$$
\operatorname{sign}(\widetilde K)=-\operatorname{sign}(\kappa),\quad
\operatorname{sign}(G_8)=\operatorname{sign}( \mu_2)=\operatorname{sign}(F_2).
$$
\end{remark}

We observe that for   systems \eqref{sys:14a)-d=1,f=0}  the same
relations between the signs of invariant polynomials as in the
case of systems
\eqref{sys:14a)-d=f=1} hold. Moreover as we shall use the same Table 1
of \cite{Art-Llib-Vlp-IJBCh}  and the same conditions for infinite
singularities (i.e. Lemma \ref{lem:ISP-mf=2}), we only need to
detect if for systems \eqref{sys:14a)-d=1,f=0} we could obtain
some configurations, which we have not obtained for systems
\eqref{sys:14a)-d=f=1}.

For this goal we only need to detect if some logically possible
configurations of singularities in the case $\mathcal{T}_4\ne0$
could not be realized for   systems \eqref{sys:14a)-d=f=1}. And then
to examine the respective case for systems \eqref{sys:14a)-d=1,f=0}
and to find out if such detected configuration could be realized for
systems \eqref{sys:14a)-d=1,f=0}.
\smallskip

(1) {\it The possibility $\kappa<0$.} We observe that all
the logically possible configurations for systems \eqref{sys:14a} in
the case $\kappa<0$ and $\mathcal{T}_4\ne0$ are realized. More
precisely we have the following number  of configurations in the
mentioned case
\begin{gather*}
\mu_2<0,\; W_4<0   \Rightarrow \quad 3; \qquad
\mu_2<0,\; W_4>0   \Rightarrow \quad 6; \\
\mu_2<0,\; W_4=0   \Rightarrow \quad 9; \qquad
\mu_2>0,\; W_4<0   \Rightarrow \quad 3; \\
\mu_2>0,\; W_4>0   \Rightarrow \quad 3;\qquad
\mu_2>0,\; W_4=0   \Rightarrow \quad 3.
\end{gather*}



(2) {\it The possibility $\kappa>0$.} In this case by
\eqref{val:14a)-d=f=1}  the condition $\kappa>0$ implies
$\eta>0$  for systems \eqref{sys:14a)-d=f=1}, i.e. the
configurations with $\eta\le0$ do not exists for these systems.

The question is: \textit{Could we obtain such configurations for
systems \eqref{sys:14a)-d=1,f=0}? }
The answer is \textit{no}, as from
\eqref{val:14a)-d=1,f=0} we observe that the condition $\kappa>0$
gives $eu<0$ and then clearly $\eta>0$.

Considering Remark \ref{rem:kappa-ne0->M-ne0-f0} and
\cite{Art-Llib-Vlp-IJBCh}  we conclude  that all   logical
possibilities for configurations in the case $\kappa>0$  and
$\mathcal{T}_4\ne0$ are realized for systems
\eqref{sys:14a)-d=f=1}.  More precisely we have the following
number  of configurations in the mentioned case
\begin{gather*}
\mu_2<0    \Rightarrow  1;\quad
\mu_2>0,\ W_4<0   \Rightarrow  1;\\
\mu_2>0,\ W_4>0   \Rightarrow \ 1; \quad
\mu_2>0,\ W_4=0   \Rightarrow  1.
\end{gather*}

Thus we  consider now the next possibility.

\subsubsection*{The subcase $\mathcal{T}_4=0$.} Then $c=0$ and we
get the systems
\begin{equation} \label{sys:14a)-d=1,fc=0}
  \dot x=  (1 + 2 u x) y,\quad \dot y  = ex -ex^2,\quad eu(1+2u)\ne 0.
\end{equation}
Moreover due to the rescaling $(x,y,t)\mapsto(x,\,|e|^{1/2}y,\,
|e|^{-1/2}t)$ we can assume $e\in\{-1,1\}$.

For these systems we calculate
\begin{equation} \label{val:14a)-d=1,fc=0}
\begin{gathered}
\mu_0=\mu_1=0,\quad \mu_2= e^2 (1 + 2 u) x^2 ,\quad
 \widetilde K=4 e u x^2,\quad \kappa= -128 e u^3, \quad
 \eta= -32 e u^3,\\
\widetilde M=16 u (3 e x^2 - 2 u y^2),\quad F_2=e^2 (1 + 2 u) x^2,\quad G_8= 2 e^2 u^2 (1 + 2 u),\\
  W_4= -256 e^6 u^4 (1 + 2 u)^3,\quad
  W_3= -128 e^5 u^5 (1 + 2 u)^2,\\
  \mathcal{T}_4=\mathcal{T}_3=\mathcal{F}_1=\mathcal{F} =0,\quad
  \mathcal{T}_2=4 e^2 u^2 (1 + 2 u),  \quad
\mathcal{B} =-2 e^2  u^4,\quad \mathcal{H} =-4 e u^3.
\end{gathered}
\end{equation}
As $\mathcal{T}_4=\mathcal{T}_3=\mathcal{F} =0$ according to
\cite{Vlp-NATMA} systems \eqref{sys:14a)-d=1,fc=0} possess two
weak singularities. Moreover, since $\mathcal{F}_1=0$ these
singularities could be only centers  and/or integrable saddles. We
observe that due to $\mu_2\widetilde K\ne0$ we have
\begin{equation} \label{signs:m2,kappa}
\begin{gathered}
\operatorname{sign}(\eta)=\operatorname{sign}(\mathcal{H})
=\operatorname{sign}(\kappa)=-\operatorname{sign}(\widetilde K),\\
 \operatorname{sign}(G_8)=\operatorname{sign}( \mu_2)
=\operatorname{sign}(\mathcal{T}_2)=\operatorname{sign}(F_2).
\end{gathered}
\end{equation}

(1) {\it The possibility $\kappa<0$.} In this case we get
 $\eta<0$ and $\mathcal{H}<0$.
\smallskip

(a) {\it The case  $\mu_2<0$.} This implies $\mathcal{T}_2<0$ and
as $\mathcal{H}<0$ and $\mathcal{B}<0$, according to
\cite{Vlp-NATMA} we have two centers.
 Therefore   considering  Lemma \ref{lem:ISP-mf=2}  we get the
 global configuration  of singularities
\[
 c, c;\, \overline{\binom{2}{1}} S,
\copyright,\copyright :  \text{ Example } \Rightarrow (e=-1,\; u=-1).
\]

(b) {\it  The case  $\mu_2>0$.} Then $\mathcal{T}_2>0$ and as
$\mathcal{B}<0$, according to \cite{Vlp-NATMA} we have one saddle
and one center. At infinity we have the same configuration and we
get
\[
{\scriptstyle\$}, c;\, \overline{\binom{2}{1}} N, \copyright,\copyright :  \text{ Example }
\Rightarrow (e= 1,\;  u= 1).
\]

(2) {\it The possibility $\kappa>0$.} In this case by
\eqref{signs:m2,kappa} we have $\eta>0$ and $\mathcal{H}>0$.
\smallskip

(a) {\it  The case  $\mu_2<0$.} Then $\mathcal{T}_2<0$ and since
$\mathcal{H}>0$ and $\mathcal{B}<0$, according to \cite{Vlp-NATMA}
we have two integrable saddles. On the other hand considering the
signs of the invariant polynomials $\mu_2$, $\kappa$ and $\eta$
according to  Lemma \ref{lem:ISP-mf=2}  we get the  global
configuration of singularities
\[
 {\scriptstyle\$}, {\scriptstyle\$};\, \overline{\binom{2}{1}} N, N^f,N^f
:  \text{ Example } \Rightarrow (e= 1,\; u=-1).
\]

(b) {\it  The case  $\mu_2>0$.} Then   we obtain
$\mathcal{T}_2>0$ and as $\mathcal{B}<0$, according to
\cite{Vlp-NATMA} we have one saddle and one center. So considering
Lemma \ref{lem:ISP-mf=2}  we get the  configuration
\[
 {\scriptstyle\$}, c;\, \overline{\binom{2}{1}} S, N^f,N^f :
 \text{ Example } \Rightarrow (e= -1,\; u= 1).
\]

Thus in the case   $\kappa\ne0$ all  the possibilities are
examined for systems \eqref{sys:14a}.


We observe that the four configurations  detected for systems
\eqref{sys:14a)-d=1,fc=0} are not realizable for systems
\eqref{sys:14a)-d=f=1}.  In order to insert these configurations
in the global diagram we use the next remark.

\begin{remark} \label{rem:insert} \rm
For the four configurations above the following conditions are
satisfied, respectively:
\begin{gather*}
c, c;\, \overline{\binom{2}{1}} S,
\copyright,\copyright\quad  \Rightarrow\quad \kappa<0,\mu_2<0,W_4>0,W_3<0,
\mathcal{T}_4=\mathcal{T}_3=\mathcal{F}_1=0;\\
{\scriptstyle\$}, c;\, \overline{\binom{2}{1}} N,
\copyright,\copyright \quad  \Rightarrow\quad \kappa<0,\mu_2>0,W_4<0,
\mathcal{T}_4=\mathcal{T}_3=\mathcal{F}_1=0;\\
{\scriptstyle\$}, {\scriptstyle\$};\, \overline{\binom{2}{1}} N, N^f,N^f \quad
\Rightarrow\quad \kappa>0,\mu_2<0,W_4>0,W_3>0,
\mathcal{T}_4=\mathcal{T}_3=\mathcal{F}_1=0;\\
{\scriptstyle\$}, c;\, \overline{\binom{2}{1}}S, N^f,N^f\quad
\Rightarrow\quad \kappa>0,\mu_2>0,W_4<0,
\mathcal{T}_4=\mathcal{T}_3=\mathcal{F}_1=0;
\end{gather*}
\end{remark}


\subsubsection*{The case $\kappa=0$.} Considering \eqref{val:14a} due to
$\widetilde K\ne0$ we obtain $d=0$ and then by Remark
\ref{rem:14a-d=1,0} we may assume $f=1$. Thus we arrive at the
following systems
\begin{equation} \label{sys:14a)-d=0}
\begin{gathered}
 \dot x =  cx(1-x),\quad cu(1+2u)\ne0,\\
\dot y  = e x  + y- e x^2 + 2 u x y,
\end{gathered}
\end{equation}
for which due to the rescaling $(x,y,t)\mapsto(x,\,ey,\, t)$ (if $e\ne0$)
 we can assume $e\in\{0,1\}$.
For these systems we calculate
\begin{equation} \label{val:14a)-d=0}
\begin{gathered}
\mu_0=\mu_1=\kappa=\kappa_1=0,\quad \mu_2= c^2 (1 + 2 u) x^2,\quad
 \widetilde K=-4 c u x^2, \\
 \tilde L=8 c (c + 2 u) x^2,\quad   K_2=96 c^2 (c^2 + 3 c u + 4 u^2) x^2,\\
\eta=0,\quad \widetilde M=-8 (c + 2 u)^2 x^2,\quad
 C_2=e x^3 - (c + 2 u)x^2 y,\quad  G_8= 0,\\
 F_2=c^2 (1 + 2 u) x^2,\quad \mathcal{T}_i=0,\ (i=1,2,3,4),\quad
 \sigma=1 + c - 2 (c - u) x,\\
\mathcal{F}_1 = \mathcal{H} =\mathcal{B}   =\mathcal{B}_1
=\mathcal{B}_2 =0,\quad \mathcal{B}_3
= 72 c^2 (1 + c) ( 1 - c + 2 u) x^2\equiv 72c^2\rho_1\rho_2x^2.
\end{gathered}
\end{equation}
Considering the  values of the above invariant polynomials
according to \cite{Vlp-NATMA} (see the Main Theorem) we arrive at
the following remark.

\begin{remark} \label{rem:is <=> B3=0} \rm
Systems \eqref{sys:14a)-d=0} possess at least one weak singularity if and
only if $\mathcal{B}_3=0$. More exactly  as $\widetilde K\ne0$, by
\cite{Vlp-NATMA} we have one integrable saddle in the case
$\sigma\ne0$ (the statement $(e_3)[\gamma]$), and we have two
integrable saddles in the case $\sigma=0$ (the statement
$(f_4)[\gamma]$).
\end{remark}

\subsubsection*{The subcase $\widetilde K<0$.} As $G_8=0$
according to \cite{Art-Llib-Vlp-IJBCh} (see Table 1, lines 148, 150)
we have two saddles if $F_2<0$ and a saddle and a node if $F_2>0$.
On the other hand from \eqref{val:14a)-d=0} it follows
$\operatorname{sign}(F_2)=\operatorname{sign}(\mu_2)$.
\smallskip

(1) {\it The possibility $\mu_2<0$.} Then $F_2<0$ and
systems \eqref{val:14a)-d=0} possess two saddles.

On the other hand by \eqref{val:14a)-d=0} the condition $\widetilde
K<0$ gives $cu>0$ and then   $\widetilde M\ne0$. Therefore   as
$\widetilde K<0$ and $\mu_2<0$ according to Lemma \ref{lem:ISP-mf=2}
and Remark \ref{rem:is <=> B3=0}   we arrive at the following three
global configurations of singularities:
\begin{gather*}
\begin{aligned}
 s, s ;\, \binom{2}{2}\,
  \overset{{}_\curvearrowright}PE\overset{
  {}_\curvearrowleft}P-\overset{
  {}_\curvearrowright}PE\overset{{}_\curvearrowleft}P,\,N^f:
 \text{ Example } \Rightarrow &(c= -2 ,\; e=0,\; u=-1) \\
&(\text{if }\sigma \mathcal{B}_3\ne0);
\end{aligned} \\
\begin{aligned}
{\scriptstyle\$}, s ;\, \binom{2}{2}
  \overset{{}_\curvearrowright}PE\overset{
  {}_\curvearrowleft}P-\overset{
  {}_\curvearrowright}PE\overset{{}_\curvearrowleft}P,\,N^f:
 \text{ Example } \Rightarrow &(c= -1 ,\; e=0,\; u=-2) \\
&(\text{if }\sigma\ne0,\; \mathcal{B}_3=0);
\end{aligned}\\
 {\scriptstyle\$}, {\scriptstyle\$} ;\, \binom{2}{2}\,
  \overset{{}_\curvearrowright}PE\overset{
  {}_\curvearrowleft}P-\overset{
  {}_\curvearrowright}PE\overset{{}_\curvearrowleft}P,\,N^f:
 \text{ Example } \Rightarrow (c= -1 ,\; e=0,\; u=-1) \quad
(\text{if }\sigma=0).
\end{gather*}

(2) {\it The possibility $\mu_2>0$.} Then $F_2>0$ and
systems \eqref{val:14a)-d=0} possess one  saddle and one node. We observe
that the Jacobian matrices for the singularities $M_1(0,0)$ and
$M_2(1,0)$ are:
\[
 M_1 \ \Rightarrow\
\begin{pmatrix} c & 0\\ e & 1 \end{pmatrix};\qquad
 M_2 \ \Rightarrow\
\begin{pmatrix} -c & 0\\ -e &  1+2u \end{pmatrix}.
\]
Therefore systems \eqref{sys:14a)-d=0} possess a node with
coinciding eigenvalues if and only if $(c-1)(c+2u+1)=0$, and this
node is a star node if and only if $(c-1)(c+2u+1)=e=0$.

On the other hand for these systems we have
$W_{11}=  96  c u^3 ( 1 - c)(1 + c + 2 u) x^4$ and
$$
 U_3\big|_{c=1}=-24 e u (1 + u)^2 x^5, \quad
U_3\big|_{c=-1-2u}=24 e u (1 + u)^2 (1 + 2 u) x^5.
$$
Since the conditions $\widetilde K\ne0$ and $\mu_2>0$ imply
$cu (1+ u)(1 + 2 u)\ne0$ we have the next remark.

 \begin{remark} \label{rem:nd,n*} \rm
Systems \eqref{sys:14a)-d=0} with $\mu_2>0$ possess a node with
coinciding eigenvalues if and only if $W_{11}= 0$. Moreover this
node is $n^d$ if $U_3\ne0$ and it is a star node if $U_3=0$.
\end{remark}

(a) {\it The case $W_{11}\ne0$.} Then the node is generic. We
observe that in this case $\sigma\ne0$, otherwise we get $c=u=-1$
and this contradicts $\mu_2>0$.
\smallskip

(a.1) {\it The subcase $\mathcal{B}_3\ne0$.} Then by Remark
\ref{rem:is <=> B3=0} the saddle is strong and we shall examine the
infinite singularities. We have again $\widetilde M\ne0$ (due to
$cu>0$) and considering \eqref{val:14a)-d=0} we obtain $K_2>0$. So
according to  Lemma \ref{lem:ISP-mf=2}  we get the configuration
\[
s, n ;\,\binom{2}{2}\,\overset{{}_\curvearrowright}P\overset{
  {}_\curvearrowleft}PH-\overset{
  {}_\curvearrowright}P\overset{{}_\curvearrowleft}PH,\,N^f:
 \text{ Example } \Rightarrow (c=  2 ,\; e=0,\; u= 1).
\]

(a.2)  {\it The subcase $\mathcal{B}_3=0$.} Since $\sigma\ne0$, by
Remark \ref{rem:is <=> B3=0} we have an integrable saddle and we
arrive at the  global configuration
\[
{\scriptstyle\$}, n ;\,\binom{2}{2}\,\overset{{}_\curvearrowright}P\overset{
  {}_\curvearrowleft}PH-\overset{
  {}_\curvearrowright}P\overset{{}_\curvearrowleft}PH,\,N^f:
 \text{ Example }  (c=  -1 ,\; e=0,\; u= -1/3).
\]

(b) {\it The case $W_{11}=0$.} Then one of the finite
singularities is a node with coinciding eigenvalues. Due to the
Remark \ref{rem:CS-(0,0)} without loss of generality we may assume
that such a node is $M_1(0,0)$, i.e. the condition $c=1$ holds.
\smallskip

(b.1) {\it The subcase $ U_3\ne0$.} Then $e\ne0$ and by Remark
\ref{rem:nd,n*} besides the saddle we have a node $n^d$. On the
other hand if $c=1$ then $u>0$ (due to $\widetilde K<0$) and we
obtain $\mathcal{B}_3\ne0$, i.e. the saddle is strong. Thus   we
obtain  the configuration
\[
s, n^d ;\,\binom{2}{2}\,\overset{
{}_\curvearrowright}P\overset{
  {}_\curvearrowleft}PH-\overset{
  {}_\curvearrowright}P\overset{{}_\curvearrowleft}PH,\,N^f:
 \text{ Example }  (c=  1 ,\; e=1,\; u= 1).
\]

(b.2) {\it The subcase $ U_3=0$.} In this case we have a star
node and a strong saddle and this leads to the   global
configuration  of singularities
\[
s, n^* ;\,\binom{2}{2}\,\overset{{}_\curvearrowright}P\overset{
  {}_\curvearrowleft}PH-\overset{
  {}_\curvearrowright}P\overset{{}_\curvearrowleft}PH,\,N^f:
 \text{ Example } (c=  1 ,\; e=0,\; u= 1) .
\]


\subsubsection*{The subcase $\widetilde K>0$.}  Since $G_8=0$
according to \cite{Art-Llib-Vlp-IJBCh} the types of the finite
singularities of systems \eqref{sys:14a)-d=0} are governed by the
polynomial $F_2$.
\smallskip

(1) {\it The possibility $\mu_2<0$.} Then $F_2<0$ and as
$G_8=0$ according to \cite{Art-Llib-Vlp-IJBCh} (see Table 1, line
164) systems \eqref{sys:14a)-d=0} possess two nodes.
\smallskip

(a) {\it The case $\widetilde M\ne0$.} Then at infinity we have
two real distinct singularities.
\smallskip

(a.1) {\it The subcase $W_{11}\ne0$.} Then  by Remark
\ref{rem:nd,n*} both nodes are generic. On the other hand
according to
 Lemma \ref{lem:ISP-mf=2}  the configuration of the infinite
singularities depends  on the sign of the invariant polynomial
$\tilde L$ (we note that  $\tilde L\ne0$ due to $\widetilde M\ne0$).
So we get the following two  global configurations of singularities
\begin{gather*}
 n, n ;\,\binom{2}{2}\,\overset{
{}_\curvearrowleft}PH\overset{
  {}_\curvearrowright}P-\overset{
  {}_\curvearrowleft}PH\overset{{}_\curvearrowright}P, S:
\text{ Example } \Rightarrow (c=  2 ,\; e=0,\; u=-2)
\quad (\text{if }\tilde L<0);
\\
\begin{aligned}
n, n ;\,\binom{2}{2}\,HHH-HHH,\,N^\infty:
\text{ Example } \Rightarrow &(c=  2 ,\; e=0,\; u=-2/3) \\
&(\text{if }\tilde L>0).
\end{aligned}
\end{gather*}

(a.2) {\it The subcase $W_{11}=0$.} Then we may assume $c=1$ and
hence the singular point $M_1$  of systems
\eqref{sys:14a)-d=0}  is a node with coinciding
eigenvalues, whereas for the second singularity $M_2$ we have
$\lambda_1=-1$ and $\lambda_2=2u+1$. Therefore the second node
will be a node with coinciding eigenvalues if and only if $u=-1$.
For $c=1$ for systems \eqref{sys:14a)-d=0} we calculate
\begin{equation} \label{val:U1,U3,mu2,L}
\begin{gathered}
U_1=-4 u (1 + u) x^2,\quad U_3=-24 e u (1 + u)^2 x^5,\quad \mu_2=(1 + 2 u) x^2,\\
\widetilde K=-4 u x^2,\quad U_5\big|_{u=-1}=-6 e x^2,\quad
 \tilde L=8(2u+1)
\end{gathered}
\end{equation}
and due to $\widetilde K\ne0$ the condition $u=-1$ is equivalent
to $U_1=0$. Moreover if $U_1\ne0$ the condition $e=0$ (to have a
star node) is equivalent to $U_3=0$. In the case $U_1=0$ (i.e.
$u=-1$) the condition $e=0$ is equivalent to $U_5=0$ and in this
case we have two star nodes.

On the other hand   due to $\mu_2\widetilde K\ne0$ we get $\tilde
L<0$. Thus considering Lemma \ref{lem:ISP-mf=2} we arrive at the
following four  global configurations of singularities
\begin{gather*}
\begin{aligned}
n, n^d ;\,\binom{2}{2}\overset{
{}_\curvearrowleft}PH\overset{
  {}_\curvearrowright}P-\overset{
  {}_\curvearrowleft}PH\overset{{}_\curvearrowright}P,S:
\text{ Example } \Rightarrow &(c=  1 ,\; e=1,\; u=-2) \\
&(\text{if } U_1\ne 0,\; U_3\ne0);
\end{aligned}\\
\begin{aligned}
n, n^* ;\,\binom{2}{2}\overset{
{}_\curvearrowleft}PH\overset{
  {}_\curvearrowright}P-\overset{
  {}_\curvearrowleft}PH\overset{{}_\curvearrowright}P,S:
\text{ Example } \Rightarrow &(c=  1 ,\; e=0,\; u=-2) \\
&(\text{if }U_1\ne 0,\; U_3=0);
\end{aligned}\\
\begin{aligned}
 n^d, n^d ;\,\binom{2}{2}\,\overset{
{}_\curvearrowleft}PH\overset{
  {}_\curvearrowright}P-\overset{
  {}_\curvearrowleft}PH\overset{{}_\curvearrowright}P,S:
\text{ Example } \Rightarrow &(c=  1 ,\; e=1,\; u=-1) \\
&(\text{if }U_1= 0,\; U_5\ne0);
\end{aligned}\\
\begin{aligned}
 n^*, n^* ;\,\binom{2}{2}\,\overset{
{}_\curvearrowleft}PH\overset{
  {}_\curvearrowright}P-\overset{
  {}_\curvearrowleft}PH\overset{{}_\curvearrowright}P,S:
\text{ Example } \Rightarrow &(c=  1 ,\; e=0,\; u=-1) \\
& (\text{if } U_1= 0,\; U_5=0).
\end{aligned}
\end{gather*}

(b) {\it The case $\widetilde M=0$.} Then we have $c=-2u$ and
this gives $C_2=e x^3$ and $W_{11}=-192 u^4 (1 + 2 u) x^4\ne0$
(due to $\mu_2\ne0$). Therefore   at infinity we have one real
singularity of multiplicity five if $C_2\ne0$ and the infinite
line is filled up with singularities if $C_2=0$. At the same time
due to Remark \ref{rem:nd,n*} systems \eqref{sys:14a)-d=0} possess
two generic nodes.
\smallskip

(b.1) {\it The subcase $C_2\ne0$.}  As $\widetilde K\ne0$ and
$\mu_2<0$ according to  Lemma \ref{lem:ISP-mf=2}    we arrive at the
global configuration  of singularities
\[
n, n ;\,\binom{2}{3}\,HH\overset{{}_\curvearrowright}P-\overset{
{}_\curvearrowleft}PHH:  \text{ Example } \Rightarrow
(c= 2,\; e=1,\; u=-1).
\]

(b.2) {\it The subcase $C_2=0$.} Then we have $c+2u=e=0$ and as
the nodes are generic, considering  Lemma \ref{lem:ISP-mf=2}   we
obtain the configuration
\[
n, n ;\,[\infty;\, S]:
\text{ Example } \Rightarrow (c=  2 ,\; e=0,\; u=-1).
\]

(2) {\it The possibility $\mu_2>0$.} Then $F_2>0$ and as
$G_8=0$ according to \cite{Art-Llib-Vlp-IJBCh} (see Table 1, line
150) systems \eqref{val:14a)-d=0} possess a saddle and a node. We
observe that due to $\widetilde K>0$ and  $\mu_2>0$ in this case
the conditions $2u+1>0$ and $cu<0$ hold.
\smallskip

(a)  {\it The case $\widetilde M\ne0$.} Then at infinity we have
two real distinct singularities.
\smallskip

(a.1)  {\it The subcase  $\mathcal{B}_3\ne0$.} Then by Remark
\ref{rem:is <=> B3=0} the saddle is strong.
\smallskip

($\alpha$)  {\it The possibility $W_{11}\ne0$.} In this
case the node is generic and as $\widetilde K\ne0$, $K_2>0$ and
$\mu_2>0$ considering    Lemma \ref{lem:ISP-mf=2}   we arrive at
the following two  global configurations of singularities
\begin{gather*}
s, n ;\,\binom{2}{2}\,\overset{
  {}_\curvearrowleft}P\overset{{}_\curvearrowright}PE-\overset{
  {}_\curvearrowleft}P\overset{{}_\curvearrowright}PE,S:
 \text{ Example } \Rightarrow (c=  -2 ,\; e=0,\; u= 2) \quad
(\text{if }\tilde L<0);
\\
s, n ;\,\binom{2}{2}\,\overset{
  {}_\curvearrowright}P\overset{{}_\curvearrowleft}PH-\overset{
  {}_\curvearrowright}P\overset{{}_\curvearrowleft}PH,\,N^\infty:
\text{ Example } \Rightarrow (c=  -2 ,\; e=0,\; u=1/4) \quad
(\text{if }\tilde L>0).
\end{gather*}

($\beta$)  {\it The possibility $W_{11}=0$.} Then we may
assume $c=1$ and considering \eqref{val:U1,U3,mu2,L}   the
condition $\mu_2>0$ implies $\tilde L=8 (1 + 2 u) x^2>0$. Moreover
in this case the condition $e=0$ is equivalent to $U_3=0$. As a
result we get the following two global configurations of
singularities:
\begin{gather*}
\begin{aligned}
 s, n^d ;\,\binom{2}{2}\,\overset{
  {}_\curvearrowright}P\overset{{}_\curvearrowleft}PH-\overset{
  {}_\curvearrowright}P\overset{{}_\curvearrowleft}PH,\,N^\infty:
\text{ Example } \Rightarrow &(c=  1 ,\; e=1,\; u=-1/4) \\
&(\text{if }U_3\ne0);
\end{aligned}\\
\begin{aligned}
 s, n^* ;\,\binom{2}{2}\,\overset{
  {}_\curvearrowright}P\overset{{}_\curvearrowleft}PH-\overset{
  {}_\curvearrowright}P\overset{{}_\curvearrowleft}PH,\,N^\infty:
\text{ Example } \Rightarrow &(c=  1 ,\; e=0,\; u=-1/4) \\
&(\text{if }U_3=0).
\end{aligned}
\end{gather*}

(a.2)  {\it The subcase  $\mathcal{B}_3=0$.} By Remark
\ref{rem:is <=> B3=0} we have an integrable saddle and we may
assume that it is located at $M_1(0,0)$, i.e. $c=-1$. Then we have
$W_{11}=-384 u^4 x^4\ne0$  and by Remark \ref{rem:nd,n*} the node
is generic. Considering  Lemma \ref{lem:ISP-mf=2}  we arrive at
the following two  global configurations of singularities:
\begin{gather*}
\begin{aligned}
{\scriptstyle \$}, n ;\,\binom{2}{2}\,\overset{
  {}_\curvearrowleft}P\overset{{}_\curvearrowright}PE-\overset{
  {}_\curvearrowleft}P\overset{{}_\curvearrowright}PE,S:
 \text{ Example } \Rightarrow &(c=  -1 ,\; e=0,\; u= 2) \\
&(\text{if }\tilde L<0);
\end{aligned}\\
\begin{aligned}
{\scriptstyle \$}, n ;\,\binom{2}{2}\,\overset{
  {}_\curvearrowright}P\overset{{}_\curvearrowleft}PH-\overset{
  {}_\curvearrowright}P\overset{{}_\curvearrowleft}PH,\,N^\infty:
\text{ Example } \Rightarrow &(c=  -1 ,\; e=0,\; u=1/4) \\
&(\text{if }\tilde L>0).
\end{aligned}
\end{gather*}

(b) {\it The case $\widetilde M=0$.} Then we have $c=-2u$ and we
obtain
\[
C_2=e x^3,\quad \mathcal{B}_3=-288 u^2 (-1 + 2 u) (1 + 4 u) x^2,\quad
W_{11}=-192 u^4 (1 + 2 u) x^4.
\]

(b.1) {\it The subcase $C_2\ne0$.} Due to $\mu_2>0$ we have
$u>-1/2$ and this implies $W_{11}\ne0$, i.e. the node is generic.
So since $\widetilde K\ne0$, $\mu_2>0$ and $K_2>0$ according to
 Lemma \ref{lem:ISP-mf=2}    we arrive at the following two
 global configurations of singularities:
\begin{gather*}
s, n  ;\,\binom{2}{3} H\overset{
{}_\curvearrowright}PE-\overset{
{}_\curvearrowleft}P\overset{{}_\curvearrowright}P\overset{
{}_\curvearrowleft}P:  \text{ Example } \Rightarrow
(c=  -2,\; e=1,\; u=1) \quad (\text{if }\mathcal{B}_3\ne0);
\\
{\scriptstyle\$}, n  ;\,\binom{2}{3}\, H\overset{
{}_\curvearrowright}PE-\overset{
{}_\curvearrowleft}P\overset{{}_\curvearrowright}P\overset{
{}_\curvearrowleft}P:  \text{ Example } \Rightarrow
(c=  -1,\; e=1,\; u= 1/2) \quad (\text{if }\mathcal{B}_3=0).
\end{gather*}

(b.2)  {\it The subcase $C_2=0$.} Then we have $c+2u=e=0$ and as
the node is generic, considering Lemma \ref{lem:ISP-mf=2}    we
obtain the configurations
\begin{gather*}
s, n  ;\,[\infty;\, N]:  \text{ Example }
\Rightarrow (c=  -2 ,\; e=0,\; u=1) \quad (\text{if }
\mathcal{B}_3\ne0);
\\
{\scriptstyle\$}, n  ;\,[\infty;\, N]:  \text{ Example }
\Rightarrow (c=  -1 ,\; e=0,\; u= 1/2) \quad (\text{if }
\mathcal{B}_3=0).
\end{gather*}


\subsubsection{Systems with $\widetilde K=0$.}
Since
$\widetilde K(\tilde a,x,y)= \operatorname{Jacob}\big(p_2(\tilde
a,x,y),q_2(\tilde a,x,y)\big)$ (see Section \ref{sec:Invariant
polyn.}) the condition $\widetilde K=0$ means that the homogeneous
quadratic parts of generic quadratic  systems are proportional,
say $q_2=\lambda p_2$. Therefore clearly the transformation
$x_1=x, y_1=-\lambda x+y$ leads to quadratic systems, of which the
second equation is linear. Applying a translation (as these
systems must have two finite real distinct singularities) we get
the systems
$$
\dot x =cx+dy  + g x^2  + 2 h x y + k y^2,\quad
\dot y  = ex + fy,
$$
for which we calculate
$$
\mathbf{U}=\alpha^2 (e x + f y)^2 (g x^2 + 2 h x y + k y^2)^2, \quad
\mu_2=\beta (g x^2 + 2 h x y + k y^2),
$$
where $\alpha =cf-d e$ and $\beta=f^2 g - 2 e f h + e^2 k$.  These
systems possess the singularities   $M_1(0,0)$ and
$M_2\left(-f \alpha/\beta,\,e\alpha/\beta\right)$ which are distinct due to
$\mathbf{U}>0$ and $\mu_2\ne0$. We observe that the condition $e=0$
implies the existence of the invariant line $y=0$ for these
systems. So we consider two cases: $e\ne0$ and $e=0$.

 In the first case we apply the transformation
$x_1=ex+fy$, $y_1=\beta y/(e\alpha)$ and $t_1=\beta t/(e\alpha)$
which places the point $M_2$ at the point $(0,1)$. This leads to
the family  of systems (we keep the old variables)
\begin{equation} \label{CS:S-e1}
  \dot x =c x+d y + g x^2  + 2 h x y -d y^2,\quad
 \dot y= x.
\end{equation}

If $e=0$ then $cfg\ne0$ (as $\alpha\beta\ne0$ and after the
rescaling $(x,y,t)\mapsto(-cx/g,y,t/f)$ (which replaces
$M_2(-c/g,0)$ to the point $(1,0)$) we arrive at the family of
systems
\begin{equation} \label{CS:S-e0}
 \dot x =c x+d y -c x^2  + 2 h x y +k y^2,\quad
 \dot y= y.
\end{equation}
In what follows we consider each one of the  families of
systems we obtained.

\subsection*{A. Systems \eqref{CS:S-e1}} We observe
that the Jacobian matrices for the singularities $M_1(0,0)$ and
$M_2(0,1)$ of these systems are respectively
$\begin{pmatrix} c & d\\ 1 &0 \end{pmatrix}$ and
$\begin{pmatrix} c+2h & -d\\ 1 & 0\end{pmatrix}$.
So the next remark becomes obvious.

\begin{remark} \label{rem:S-e1-no n*} \rm
The family of systems \eqref{CS:S-e1} could not have a finite star
node.
\end{remark}

For systems \eqref{CS:S-e1} we calculate
\begin{equation} \label{val:S-e1}
\begin{gathered}
\mu_0=\mu_1=\kappa=\widetilde K=0,\quad \eta=4 g^2 (d g + h^2),\quad
\mu_2=-d (g x^2 + 2 h x y - d y^2),\\
 \theta_2=d g + h^2,\quad \mathbf{U}= d^2 x^2 (g x^2 + 2 h x y - d y^2)^2,
\quad G_8= 2 d g (d g + h^2),\\
  W_4= 16 d^2 g^2(d g + h^2)^2  (c^2 + 4 d)(c^2 - 4 d + 4 c h + 4 h^2)=
  16 d^2 g^2(d g + h^2)^2 \tau_1\tau_2,\\
\mathbf{D}=-192 d^4 (d g + h^2), \quad \tilde L=8 g (g x^2 + 2 h x y - d y^2),\\
\mathcal{F}_1=2 d (c g + h + g h), \quad
\mathcal{T}_4=4 c d g (c + 2 h) (d g + h^2).
\end{gathered}
\end{equation}

\begin{remark} \label{rem:W_4-ne0=>G8,D,theta2-ne} \rm
The condition  $W_4\ne0$  implies $G_8\mathbf{D}\theta_2\eta\ne0$ for
systems \eqref{CS:S-e1}.
\end{remark}


\subsubsection*{The case $W_4<0$.} Then by the above remark we have $G_8\mathbf{D}\ne0$
and due to $\widetilde K=0$, according to \cite{Art-Llib-Vlp-IJBCh} (see
Table 1, lines 157,162)  we have a saddle and either a focus or a
center.

\subsubsection*{The subcase $\mathcal{T}_4\ne0$.} By
\cite{Vlp-NATMA} we have a strong saddle and a strong focus. On
the other hand as by Remark \ref{rem:W_4-ne0=>G8,D,theta2-ne} the
condition   $\eta\theta_2\ne0$ holds, considering Lemma
\ref{lem:ISP-mf=2} we obtain the following three global
configurations of singularities
\begin{gather*}
\begin{aligned}
s, f ;\,N^d,\,\binom{1}{1}\,\copyright,\binom{1}{1}\,
\copyright:  \text{ Example } \Rightarrow &(c=1,\; d=-2,\; g=1,\; h=1) \\
& (\text{if }\eta<0);
\end{aligned}\\
\begin{aligned}
s, f ;\,\overline{\binom{1}{1}}SN,\,\overline{\binom{1}{1}}SN,\,N^d:  \text{ Example }
\Rightarrow &(c=1,\;d=-2,\; g=-1,\; h=1) \\
&(\text{if }\eta>0,\;
\mu_2\tilde L<0);
\end{aligned}\\
\begin{aligned}
s, f ;\,\overline{\binom{1}{1}}
SN, \overline{\binom{1}{1}} NS,\,N^d:  \text{ Example }
\Rightarrow &(c=1,\;d=3,\; g=-1/4,\; h=1) \\
&(\text{if }\eta>0\; \mu_2\tilde L>0).
\end{aligned}
\end{gather*}

\subsubsection*{The subcase $\mathcal{T}_4=0$.}
Then one of the finite singularities is weak and by Remark \ref{rem:CS-(0,0)}
without loss of generality we may assume $\rho_1=c=0$, i.e. this
weak singularity is $M_1(0,0)$. In this case for systems
\eqref{CS:S-e1} we calculate:
\begin{equation} \label{val:T3,F,F1-S1}
\begin{gathered}
\mathcal{T}_3=8 d g h (d g + h^2),\quad
\mathcal{F}=g h (d g + h^2),\quad \mathcal{F}_1=2 d h (1 + g),\\
\mathcal{F}_2=\mathcal{F}_3=0,\quad W_4=-256 d^3 g^2 (d - h^2) (d g + h^2)^2.
\end{gathered}
\end{equation}

(a) {\it The possibility $\mathcal{T}_3\mathcal{F}<0$.} According
to \cite{Vlp-NATMA} the systems possess a weak focus, the order of
which is determined by the invariant polynomial $\mathcal{F}_1$.
\smallskip

(a.1) {\it  The case $\mathcal{F}_1\ne0$.} Then we have a first
order weak focus and considering   Lemma \ref{lem:ISP-mf=2}  and
the conditions $\eta\theta_2\ne0$ and $\kappa=0$ we obtain the
following three  global configurations of singularities:
\begin{gather*}
\begin{aligned}
s, f^{(1)} ;\,N^d,\,\binom{1}{1}\,\copyright,\binom{1}{1}\, \copyright:  \text{ Example } \Rightarrow
&(c=0,\;d=-1,\; g=2,\; h=1) \\ &(\text{if }\eta<0);
\end{aligned} \\
\begin{aligned}
s, f^{(1)} ;\,\overline{\binom{1}{1}}
SN, \overline{\binom{1}{1}} SN,\,N^d:  \text{ Example }
\Rightarrow &(c=0,\;d=-1,\; g=-2,\; h=1) \\
&(\text{if }\eta>0,\;
\mu_2\tilde L<0);
\end{aligned}\\
\begin{aligned}
s, f^{(1)} ;\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}} NS,\,N^d:  \text{ Example }
\Rightarrow &(c=0,\; d=-1,\; g= 1/2,\; h=1) \\
&(\text{if }\eta>0,\; \mu_2\tilde L>0).
\end{aligned}
\end{gather*}

(a.2) {\it  The case $\mathcal{F}_1=0$.} Then by
\eqref{val:T3,F,F1-S1} due to $\mathcal{T}_3\ne0$ we get $g=-1$
and then we have
\begin{gather*}
\mu_2\tilde L=8 d (x^2 - 2 h x y + d y^2)^2, \quad
\mathcal{T}_3\mathcal{F}= 8 d h^2 (d - h^2)^2,\\
\eta=4(h^2-d),\quad  W_4=256 d^3 (h^2-d)^3.
\end{gather*}
So we arrive at the following remark.

\begin{remark} \label{rem:F1=0=>mu2L*T3F>0}\rm
 If $\mathcal{T}_4=0$ and $\mathcal{T}_3\ne0$ then the condition
$\mathcal{F}_1=0$ implies
$\operatorname{sign}(\mathcal{T}_3\mathcal{F})
=\operatorname{sign}(\mu_2\tilde L)$=$\operatorname{sign}(\eta W_4)$.
\end{remark}

Since  by \eqref{val:T3,F,F1-S1} we have
$\mathcal{F}_2=\mathcal{F}_3=0$, according to \cite{Vlp-NATMA} systems
\eqref{CS:S-e1} possess a center. By the above remark  the
conditions $\mathcal{T}_3\mathcal{F}<0$ and $W_4<0$ imply
$\mu_2\tilde L<0$ and $\eta>0$. So considering Lemma
\ref{lem:ISP-mf=2} we obtain the configuration
\[
s, c;\,\overline{\binom{1}{1}} SN,\,\overline{\binom{1}{1}} SN,\,N^d:  \text{ Example } \Rightarrow
(c=0,\;d=-2,\; g= -1,\;h=1).
\]

(b) {\it The possibility $\mathcal{T}_3\mathcal{F}>0$.} In this
case by  \cite{Vlp-NATMA} the systems possess a weak saddle.
\smallskip

(b.1) {\it  The case $\mathcal{F}_1\ne0$.} Then we have a first
order weak saddle and considering   Lemma \ref{lem:ISP-mf=2}  and
the conditions $\eta\theta_2\ne0$ and $\kappa=0$ we obtain the
following three  global configurations of singularities:
\begin{gather*}
\begin{aligned}
s^{(1)}, f;\,N^d,\,\binom{1}{1}\,\copyright,\binom{1}{1} \copyright:
 \text{ Example } \Rightarrow &(c=0,\;d=2,\; g=-2,\; h=1) \\
& (\text{if }\eta<0);
\end{aligned} \\
\begin{aligned}
s^{(1)}, f;\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}} SN,\,N^d:  \text{ Example }
\Rightarrow &(c=0,\;d=2,\; g= 1,\; h=1) \\
& (\text{if }\eta>0,\; \mu_2\tilde L<0);
\end{aligned}\\
\begin{aligned}
s^{(1)}, f;\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}} NS,\,N^d:  \text{ Example }
\Rightarrow &(c=0,\;d=2,\; g= -1/3,\; h=1) \\
& (\text{if }\eta>0,\; \mu_2\tilde L>0).
\end{aligned}
\end{gather*}

(b.2) {\it  The case $\mathcal{F}_1=0$.} Then the saddle is
integrable. On the other hand  by Remark
\ref{rem:F1=0=>mu2L*T3F>0} the condition $W_4<0$ implies $\eta<0$
and this leads to the configuration
\[
{\scriptstyle\$}, f;\,N^d,\,\binom{1}{1}\,\copyright,\binom{1}{1}\,
\copyright:  \text{ Example } \Rightarrow (c=0,\;d=2,\; g=-1,\; h=1).
\]

(c) {\it The possibility $\mathcal{T}_3=0$.} By
\eqref{val:T3,F,F1-S1} due to $W_4\ne0$   the condition
$\mathcal{T}_3=0$ implies $h=0$ and then we obtain
\begin{gather*}
 \mathcal{T}_4=\mathcal{T}_3=\mathcal{F}_1=0,\quad
 \mathcal{T}_2=4 d^2 g^2,\quad  \mathcal{B}=-2 d^2 g^4, \\
 W_4=-256 d^6 g^4,\quad \eta =4 d g^3,\quad
 \mu_2\tilde L=-8 d g (g x^2 - d  y^2)^2.
\end{gather*}

\begin{remark} \label{rem:T4=T3=0=>W_4>0} \rm
If  $W_4\ne0$ then the condition
$\mathcal{T}_4= \mathcal{T}_3=0$ implies $W_4<0$.
\end{remark}

The condition $W_4\ne0$ implies $\mathcal{B}<0$ and $\mathcal{T}_2>0$.
Therefore by \cite{Vlp-NATMA} we have a center and an integrable
saddle. As the condition $\eta>0$ implies $\mu_2\tilde L<0$,
taking into consideration Lemma \ref{lem:ISP-mf=2}  we get
following two configurations:
\begin{gather*}
{\scriptstyle\$}, c;\,N^d,\,\binom{1}{1}\,\copyright,\binom{1}{1}\,
\copyright: \text{ Example } \Rightarrow (c=0,\;d=1,\; g=-1,\; h=0)
\quad (\text{if }\eta<0);
\\
{\scriptstyle\$}, c ;\,\overline{\binom{1}{1}}SN,\overline{\binom{1}{1}} SN,\,N^d: \text{ Example }
\Rightarrow (c=0,\;d=1,\; g= 1,\; h=0) \quad (\text{if }\eta>0).
\end{gather*}

\subsubsection*{The case $W_4>0$.} Since $\widetilde K=0$ and $G_8\mathbf{D}\ne0$
(see  Remark \ref{rem:W_4-ne0=>G8,D,theta2-ne}), according to
\cite{Art-Llib-Vlp-IJBCh} (see Table 1, line  151) systems
\eqref{CS:S-e1} possess   a saddle and a node.
\smallskip

\subsubsection*{The subcase $\mathcal{T}_4\ne0$.} Then by
\cite{Vlp-NATMA} the saddle is strong  and as $W_4\ne0$ the node is
generic and $\theta_2\ne0$. Therefore considering Lemma
\ref{lem:ISP-mf=2} we obtain the following three  global
configurations of singularities
\begin{gather*}
\begin{aligned}
s, n ;\,N^d,\,\binom{1}{1}\,\copyright,\binom{1}{1}\,
\copyright: \text{ Example } \Rightarrow &(c=1,\;d=1/2,\; g=
-3,\; h= 1) \\
& (\text{if }\eta<0);
\end{aligned}\\
\begin{aligned}
s, n ;\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}} SN,\,N^d: \text{ Example }
\Rightarrow &(c=1,\;d=1/2,\; g= 1,\; h= 1) \\
& (\text{if }\eta>0,\; \mu_2\tilde L<0);
\end{aligned}\\
\begin{aligned}
s, n ;\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}} NS,\,N^d: \text{ Example }
\Rightarrow &(c=1,\;d=1/2,\; g= -1,\; h= 1) \\
& (\text{if }\eta>0,\; \mu_2\tilde L>0).
\end{aligned}
\end{gather*}

\subsubsection*{The subcase $\mathcal{T}_4=0$.} Then the saddle is
weak and by Remark \ref{rem:CS-(0,0)} without loss of generality we
may assume that this saddle is located at $M_1(0,0)$, i.e. $c=0$ and
we consider the relations \eqref{val:T3,F,F1-S1}.

We observe that by Remark \ref{rem:T4=T3=0=>W_4>0} the condition
$W_4>0$ implies   $\mathcal{T}_3\ne0$.
\smallskip

(a) {\it The possibility $\mathcal{F}_1\ne0$.} In this case by
\cite{Vlp-NATMA}  the weak saddle is of order one and considering
Lemma \ref{lem:ISP-mf=2}  we arrive at the following three global
configurations of singularities:
\begin{gather*}
\begin{aligned}
s^{(1)}, n;\,N^d,\,\binom{1}{1}\,\copyright,\binom{1}{1}\, \copyright: \text{ Example } \Rightarrow &(c=0,\;d=1/2,\;
g= -3,\; h=1) \\
&(\text{if }\eta<0) ;
\end{aligned}\\
\begin{aligned}
s^{(1)}, n ;\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}} SN,\,N^d: \text{ Example }
\Rightarrow &(c=0,\;d=1/2,\; g= 1,\; h=1) \\
& (\text{if }\eta>0,\; \mu_2\tilde L<0);
\end{aligned}\\
\begin{aligned}
s^{(1)}, n ;\,\overline{\binom{1}{1}}SN,\,\overline{\binom{1}{1}} NS,\,N^d: \text{ Example }
\Rightarrow &(c=0,\;d=1/2,\; g= -1/2,\; h=1) \\
&(\text{if }\eta>0,\;\mu_2\tilde L>0).
\end{aligned}
\end{gather*}

(b) {\it The possibility $\mathcal{F}_1=0$.} As $\mathcal{F}_2=
\mathcal{F}_3=0$ according to \cite{Vlp-NATMA} systems
\eqref{CS:S-e1} possess  an integrable saddle.   On the other hand
 by Remark \ref{rem:F1=0=>mu2L*T3F>0} the condition $W_4>0$ implies $\eta>0$
and this leads to the configuration
\[
{\scriptstyle\$}, n ;\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}} NS,\,N^d: \text{ Example }
\Rightarrow (c=0,\;d=1/2,\; g= -1,\; h=1).
\]

\subsubsection*{The case $W_4=0$.} Taking into account \eqref{val:S-e1} we
consider two possibilities: $\eta\ne0$ and $\eta=0$.

\subsubsection*{The subcase $\eta\ne0$.} Then due to $\mu_2\ne0$
the condition $W_4=0$ implies $\tau_1\tau_2=(c^2 + 4 d)(c^2 - 4 d
+ 4 c h + 4 h^2)=0$, i.e. one of the singularities is a node with
coinciding eigenvalues. Moreover by Remark \ref{rem:S-e1-no n*} we
have a node with one direction. According to Remark
\ref{rem:CS-(0,0)} we may assume that the singularity $M_1(0,0)$
is such a node and this implies $ \tau_1 = c^2 + 4 d =0$, i.e.
$d=-c^2/4\ne0$. So we may assume $c=1$ due to the rescaling
$(x,y,t)\mapsto(cx,y, t/c)$ and this leads to the family of systems
\begin{equation} \label{CS:S-e1-a}
  \dot x =  x- y/4 + g x^2  + 2 h x y + y^2/4,\quad
 \dot y= x.
\end{equation}
for which we calculate
\begin{gather*}
\mu_2\tilde L=2 g  (4 g x^2 + 8 h x y + y^2)^2/8,\quad
\mathcal{T}_4=  g (1 + 2 h) (g - 4 h^2)/4,\\
 \theta_2=  (4 h^2-g)/4,\quad \eta= g^2 (4 h^2-g).
\end{gather*}

(1) {\it The possibility $\mathcal{T}_4\ne0$.} Then by
\cite{Vlp-NATMA} we have a strong saddle. We observe that the
condition $\eta\ne0$ implies $\theta_2\ne0$ and considering Lemma
\ref{lem:ISP-mf=2}  we obtain the following three
 global configurations of singularities:
\begin{gather*}
\begin{aligned}
 s, n^d ;\,N^d,\,\binom{1}{1}\,\copyright,\binom{1}{1}\,
\copyright: \text{ Example } \Rightarrow
&(c=1,\;d=-1/4,\; g= 1,\; h=0) \\
& (\text{if }\eta<0);
\end{aligned}\\
\begin{aligned}
s, n^d ;\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}}SN,\,N^d: \text{ Example }
\Rightarrow &(c=1,\;d=-1/4,\; g= -1,\; h=0) \\
&(\text{if }\eta>0,\; \mu_2\tilde L<0);
\end{aligned}\\
\begin{aligned}
s, n^d ;\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}} NS,\,N^d: \text{ Example }
\Rightarrow &(c=1,\;d=-1/4,\; g= 1,\; h=1) \\
&(\text{if }\eta>0,\; \mu_2\tilde L>0).
\end{aligned}
\end{gather*}

(2)  {\it The possibility  $\mathcal{T}_4=0$.} Then the
saddle $M_2$ is weak. As $\eta\ne0$ we get $h=-1/2$ and then
$$
\eta=  g^2(1 - g),\quad \mathcal{F}_1=(1 - g)/4,\quad
 \theta_2=(1 -g)/4.
$$
So due to $\eta\ne0$ we have $\theta_2\mathcal{F}_1\ne0$, i.e. the
weak saddle is of order one. Therefore we get the following three
configurations of singularities:
\begin{gather*}
\begin{aligned}
s^{(1)}, n^d ;\,N^d,\,\binom{1}{1}\,\copyright,\binom{1}{1}\, \copyright: \text{ Example }
\Rightarrow &(c=1,\;d=-1/4,\; g= 2,\; h=-1/2) \\
& (\text{if }\eta<0);
\end{aligned} \\
\begin{aligned}
 s^{(1)}, n^d ;\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}}SN,\,N^d: \text{ Example }
\Rightarrow &(c=1,\;d=-1/4,\; g= -1,\\
& h=-1/2) \quad (\text{if }\eta>0,\; \mu_2\tilde L<0);
\end{aligned} \\
\begin{aligned}
s^{(1)}, n^d ;\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}} NS,\,N^d: \text{ Example }
\Rightarrow &(c=1,\;d=-1/4,\; g= 1/2,\\
& h=-1/2) \quad (\text{if }\eta>0,\; \mu_2\tilde L>0).
\end{aligned}
\end{gather*}


\subsubsection*{The subcase $\eta=0$.} Considering
\eqref{val:S-e1} we have $g(d g + h^2)=0$ and we consider two
possibilities: $\tilde L \ne0$ (then $g\ne0$) and $\tilde L =0$
(then $g=0$).
\smallskip

(1) {\it The possibility $\tilde L \ne0$.} In this case
$g\ne0$ and as $d\ne0$ we obtain $g=-h^2/d\ne0$. Hence we may
assume $h=1$ due to the rescaling  $(x,y,t)\mapsto(hx,y, t/h)$ (and
by introducing new parameters $\tilde c=c/h$, $\tilde d=d/h^2$).
So we arrive at  the family of systems (we keep the old
parameters)
\begin{equation} \label{CS:S-e1-b}
  \dot x =  cx + dy - x^2/d  + 2 x y -dy^2,\quad
 \dot y= x.
\end{equation}
for which we calculate
\begin{equation} \label{val:S-e1-b}
\begin{gathered}
\kappa= \eta= \widetilde K=\mathbf{D}= 0, \quad
\kappa_1=- 32 (1 + c - d)/d,\\
\theta_5= 96(1 + c - d)(x - d y)^3)/d^2, \quad
\widetilde M=- 8 ( x - d y)^2 /d^2=-\tilde L,\\
\mu_2=(x - d y)^2,\quad  G_3=-2 (1 + c - d), \\
 W_8=2^{12}3^3(1 + c - d)^2(c^2 + 4 d)\big[(c+2)^2 - 4 d\big]
= 2^{12}3^3(1 + c - d)^2\tau_1\tau_2,\\
 \mathcal{T}_i=0,\quad i=1,2,3,4,\quad \sigma=c-2x/d+ 2y,\quad
 \mathcal{F}_1=-2 (1 + c - d),\quad \mathcal{H}=0,\\
 \mathcal{B}_1=-2 c (2 + c) (1 + c - d)=-2(1 + c - d)\rho_1\rho_2,\quad
 \mathcal{B}_2= 4 (1 + c) (1 + c - d)^2 /d.
\end{gathered}
\end{equation}

\begin{remark} \label{rem:W8-ne0=>G3, theta5-ne0} \rm
The condition  $W_8\ne0$
implies $G_3\kappa_1\theta_5 \mathcal{F}_1\ne0$ for systems
\eqref{CS:S-e1-b}. Moreover in this case the condition
$\mathcal{B}_1=0$ is equivalent to $\rho_1\rho_2=0$.
\end{remark}

Since $ \widetilde K=\mathbf{D} = 0$ according to
\cite{Art-Llib-Vlp-IJBCh} the types of the finite singularities
are governed by the invariant polynomials $W_8, W_9$, $G_3$ and
$D_2$.
\smallskip

(a) {\it The case $W_8<0$.} Then by Remark \ref{rem:W8-ne0=>G3,
theta5-ne0} we have $G_3\ne0$ and by \cite{Art-Llib-Vlp-IJBCh} (see
Table 1, line 159) systems \eqref{CS:S-e1-b} possess a saddle and a
focus.
\smallskip

(a.1) {\it The subcase $\mathcal{B}_1\ne0$.} Then
$\rho_1\rho_2\ne0$ and  we do not have weak singularities. On the
other hand as  $\widetilde M\tilde L\kappa_1\theta_5\ne0$,
considering Lemma \ref{lem:ISP-mf=2},  we obtain the global
configuration of singularities
\[
s, f ;\,\widehat{\binom{2}{2}} {\overset{
  {}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P  {H_{ \curlywedge}}-H,\,N^d: \text{ Example }
\Rightarrow (c=1,\;d=-1,\; g= 1,\; h=1).
\]

(a.2) {\it The subcase $\mathcal{B}_1=0$.} Then one of the finite
singularities is weak and by Remark \ref{rem:CS-(0,0)} without loss
of generality we may assume  such a point to be $M_1(0,0)$, i.e. for
systems \eqref{CS:S-e1-b} the condition $\rho_1=c=0$ holds. So we
have
\begin{equation} \label{val:S3;B1,B2,W8}
\mathcal{B}_1=0,\quad \mathcal{B}_2=4 (1  - d)^2 /d,\quad
 W_8= 2^{16}3^3 d (1- d)^3.
\end{equation}
Therefore the condition $W_8<0$ implies $\mathcal{B}_2\ne0$ and by
\cite{Vlp-NATMA} the type of weak singularity (which is of order
one due to $\mathcal{F}_1\ne0$) depends of the sign of
$\mathcal{B}_2$. Thus considering Lemma \ref{lem:ISP-mf=2} we get
the following two global configurations of singularities:
\begin{gather*}
\begin{aligned}
 s, f^{(1)} ;\,\widehat{\binom{2}{2}} {\overset{
  {}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P  {H_{ \curlywedge}}-H,\,N^d: \text{ Example }
\Rightarrow &(c=0,\; d=-1,\; g= 1,\; h=1)\\
&(\text{if }\mathcal{B}_2<0);
\end{aligned} \\
\begin{aligned}
s^{(1)}, f;\,\widehat{\binom{2}{2}} {\overset{
  {}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P  {H_{ \curlywedge}}-H,\,N^d: \text{ Example }
\Rightarrow &(c=0,\;d=2,\; g= -1/2,\;h=1) \\
&(\text{if }\mathcal{B}_2>0).
\end{aligned}
\end{gather*}

(b) {\it The case $W_8>0$.} Then by
\cite[Table 1, line 153]{Art-Llib-Vlp-IJBCh},
systems \eqref{CS:S-e1-b} possess   a
saddle and a node which is generic (due to $W_8\ne0$).
\smallskip

(b.1) {\it  The subcase $\mathcal{B}_1\ne0$.} Then the saddle is
strong and considering Remark \ref{rem:W8-ne0=>G3, theta5-ne0} and
Lemma \ref{lem:ISP-mf=2} we arrive at the  configuration
\[
s, n ;\,\widehat{\binom{2}{2}} {\overset{
  {}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P  {H_{ \curlywedge}}-H,\,N^d: \text{ Example }
\Rightarrow (c=1,\;d=1 ,\; g= -1,\; h=1).
\]

(b.2) {\it  The subcase $\mathcal{B}_1=0$.} It was mentioned
earlier that in this case we may assume $c=0$, i.e. the weak
saddle of the first order (as $\mathcal{F}_1\ne0$) is located at
the origin of coordinates. Considering Remark \ref{rem:W8-ne0=>G3,
theta5-ne0} and Lemma \ref{lem:ISP-mf=2} we obtain the
configuration
\[
s^{(1)}, n ;\,\widehat{\binom{2}{2}}
{\overset{  {}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P  {H_{ \curlywedge}}-H,\,N^d: \text{ Example }
\Rightarrow (c=0,\;d=1/2,\; g= -2,\; h=1).
\]

(c) {\it The case $W_8=0$.} Then  $(1 + c - d)\tau_1\tau_2=0$ and
considering \eqref{val:S-e1-b} we have to distinguish two subcases
given by the invariant polynomial $\kappa_1$.
\smallskip

(c.1) {\it The subcase $\kappa_1\ne0$.} Then $(1 + c - d)\ne0$
and the condition $W_8=0$ gives $\tau_1\tau_2=0$. Therefore we
have a node with coinciding eigenvalues and by Remark
\ref{rem:CS-(0,0)} we may assume that this node is located at the
origin  of coordinates, i.e. the condition $\tau_1=c^2 + 4
d=0$ holds. We note that by Remark \ref{rem:S-e1-no n*} this node
could not be a star node. So we have $d=-c^2/4\ne0$
 and we calculate
\[
\mathcal{B}_1=- c (2 + c)^3/2,\quad
  \kappa_1= 32 (2 + c)^2 /c^2, \quad
 \theta_5=- 6 (2 + c)^2 (4 x + c^2 y)^3/c^4.
\]
The condition $\kappa_1\ne0$ implies $\mathcal{B}_1\ne 0$  and  by
\cite{Vlp-NATMA}, the saddle is strong. On the other  hand
the condition $\kappa_1\ne0$ implies $\theta_5\ne0$ and
considering Lemma \ref{lem:ISP-mf=2} we arrive at the
configuration
\[
s , n^d ;\,\widehat{\binom{2}{2}} {\overset{
  {}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P  {H_{ \curlywedge}}-H,\,N^d: \text{ Example }
\Rightarrow (c=1,\;d=-1/4,\; g= 4,\; h=1).
\]

(c.2) {\it The subcase $\kappa_1=0$.} Then by \eqref{val:S-e1-b} we
have $c=d-1$ and for systems \eqref{CS:S-e1-b} we calculate
\begin{equation} \label{val:S-e1-b0}
\begin{gathered}
\mu_0=\mu_1=\widetilde K= \eta=\mathbf{D}=\kappa=\kappa_1=0,\quad
\widetilde M=- 8 (x - d y)^2/d^2=\mathcal{G},\\
\mu_2=(x - d y)^2,\quad  K_2= 96 (x - d y)^2 /d^2,\quad
\theta_6=  8 (x - d y)^4 /d^2,\\
G_3= 0,\quad D_2= 2(x - d y)/d,\quad \mathbf{U}=x^2 (x - d y)^4,\\
\mathcal{T}_i=0,\ i=1,2,3,4,\quad \sigma=d-1 -  2 x /d + 2 y,\quad
 \mathcal{F}_1=\mathcal{H}= 0,\\
 \mathcal{B} =\mathcal{B}_1=\mathcal{B}_2=0,\quad
\mathcal{B}_3=  72 (d-1) (1 + d) (x - d y)/d^2 .
\end{gathered}
\end{equation}
Since $\mathbf{D}=\widetilde K=G_3=0$ and $D_2\ne0$, according to
\cite{Art-Llib-Vlp-IJBCh} (see Table 1, line 154) we have a saddle
and a node. Considering \eqref{val:S-e1-b0}, by \cite{Vlp-NATMA}
(see the Main Theorem, the statement $(e_3)[\delta])$ the saddle
will be weak (more precisely it will be an integrable one) if and
only if $\mathcal{B}_3=0$. Moreover for the singular points
$M_1(0,0)$ and $M_2(0,1)$ we have, respectively
$$
\rho_1=d-1,\quad \tau_1=(d+1)^2, \quad \rho_2=d+1,\quad \tau_2=(d-1)^2.
$$
Therefore, we observe that if one of the singular points is a weak
saddle, the second one becomes a node with coinciding eigenvalues,
which by Remark \ref{rem:S-e1-no n*} is a node $n^d$.

Since $K_2>0$ and $\theta_6\ne0$,   considering Lemma
\ref{lem:ISP-mf=2}, we obtain the following  two global
configurations of singularities:
\begin{gather*}
\begin{aligned}
s, n  ;\,\binom{2}{2}\,\overset{
{}_\curvearrowright}P\overset{
  {}_\curvearrowleft}PH-\overset{
  {}_\curvearrowright}P\overset{{}_\curvearrowleft}PH,\,N^d:
\text{ Example } \Rightarrow &(c=1,\;d=2,\; g= -1/2,\; h=1)\\
&(\text{if }\mathcal{B}_3\ne0);
\end{aligned} \\
\begin{aligned}
{\scriptstyle\$}, n^d;\,\binom{2}{2}\,\overset{
{}_\curvearrowright}P\overset{
  {}_\curvearrowleft}PH-\overset{
  {}_\curvearrowright}P\overset{{}_\curvearrowleft}PH,\,N^d:
\text{ Example } \Rightarrow &(c=0,\;d=1,\; g= -1,\;h=1)\\
&(\text{if }\mathcal{B}_3=0).
\end{aligned}
\end{gather*}

(2) {\it The possibility $\tilde L =0$.} Then
$g=0$ and we obtain the family of systems
\begin{equation} \label{CS:S-e1-g0}
  \dot x =c x+d y   + 2 h x y -d y^2,\quad
 \dot y= x.
\end{equation}
for which we calculate
\begin{equation} \label{val:S-e1-g0}
\begin{gathered}
\mu_0=\mu_1= \widetilde K=\kappa= \eta=G_8= 0,\quad
 \mu_2=d y (-2 h x + d y),\quad \mathbf{D}=-192 d^4 h^2,\\
 \mathbf{U}=d^2 x^2 y^2 (2 h x - d y)^2,\quad
   W_7= 12 d^2 h^6 (c^2 + 4 d)\big[(c+2h)^2 - 4 d\big]=
 12 d^2 h^6\tau_1\tau_2,\\
\mathcal{T}_i=0,\quad i=1,2,3,4,\quad \sigma=c + 2 h y,\quad
\mathcal{F}_1=2 d h,\quad \mathcal{H}=0,\\ \mathcal{B}_1=2 c d h (c
+ 2 h)=2   d h \rho_1\rho_2,\quad \mathcal{B}_2=4 d h^3 (c + h),\quad
\widetilde M=-32 h^2 y^2.
\end{gathered}
\end{equation}

For  the singular points $M_1(0,0)$ and $M_2(0,1)$ of the above
systems we have $ \Delta_2= d=- \Delta_1$ and hence  these systems
possess a saddle and an anti-saddle.
\smallskip

(a) {\it The case $W_7<0$.}  Then we have a saddle and a focus or
a center and considering \eqref{val:S-e1-g0} we observe that the
condition $\mathcal{B}_1=0$ is equivalent to $\rho_1\rho_2=0$.
\smallskip

(a.1) {\it The subcase $\mathcal{B}_1\ne0$.} Then both
singularities are strong and considering Lemma \ref{lem:ISP-mf=2}
we arrive at the   configuration
\[
 s, f ;\,\widehat{\binom{1}{2}}{\overset{ {}_\curvearrowright}P_{\curlywedge}}E
{\overset{ {}_\curvearrowleft}P_{\curlywedge}}-
H,\,\overline{\binom{1}{1}}SN: \text{ Example }
\Rightarrow (c=1,\;d=-1,\; g=0,\ h=1).
\]

(a.2) {\it The subcase $\mathcal{B}_1=0$.}  Then one of the finite
singularities is weak and by Remark \ref{rem:CS-(0,0)} without loss
of generality we may assume  such a point is $M_1(0,0)$, i.e. for
systems \eqref{CS:S-e1-g0} the condition $\rho_1=c=0$ holds. In this
case we    calculate
\begin{equation} \label{val:S3b;B1,B2,W8}
\mathcal{B}_1=0,\quad \mathcal{B}_2= 4 d h^4,\quad \mathcal{F}_1=2dh,\quad
W_7=-192 d^3 h^6 (d - h^2).
\end{equation}
Therefore the condition $W_7\ne0$ implies
$\mathcal{F}_1\mathcal{B}_2\ne0$ and by \cite{Vlp-NATMA} the type
of the weak singularity (which is of order one due to
$\mathcal{F}_1\ne0$) depends on the sign of $\mathcal{B}_2$. Thus
considering  Lemma \ref{lem:ISP-mf=2}  we obtain the
following two global configurations of singularities:
\begin{gather*}
\begin{aligned}
s, f^{(1)} ;\,\widehat{\binom{1}{2}}{\overset{
{}_\curvearrowright}P_{\curlywedge}}E {\overset{
{}_\curvearrowleft}P_{\curlywedge}}-
H,\,\overline{\binom{1}{1}} SN: \text{ Example }
\Rightarrow &(c=0,\; d=-1,\; g=0,\; h=1)\\
& (\text{if }\mathcal{B}_2<0);
\end{aligned}\\
\begin{aligned}
s^{(1)}, f;\,\widehat{\binom{1}{2}}{\overset{
{}_\curvearrowright}P_{\curlywedge}}E {\overset{
{}_\curvearrowleft}P_{\curlywedge}}-
H,\,\overline{\binom{1}{1}}SN: \text{ Example }
\Rightarrow &(c=0,\;d=2,\; g=0,\; h=1)\\
& (\text{if }\mathcal{B}_2>0).
\end{aligned}
\end{gather*}

(b) {\it The case $W_7>0$.} Then by \cite[Table 1, line 152]{Art-Llib-Vlp-IJBCh},
systems \eqref{CS:S-e1-g0} possess  a saddle and a node which is generic
(due to $W_7\ne0$).

Taking into account the fact that the saddle is weak if and only
if $\mathcal{B}_1=0$ we   get the following two global
configurations of singularities:
\begin{gather*}
\begin{aligned}
s, n  ;\,\widehat{\binom{1}{2}}{\overset{
{}_\curvearrowright}P_{\curlywedge}}E {\overset{
{}_\curvearrowleft}P_{\curlywedge}}-
H,\,\overline{\binom{1}{1}}SN: \text{ Example }
\Rightarrow &(c=1,\;d= 1/2,\; g=0,\; h=1)\\
& (\text{if }\mathcal{B}_1\ne0);
\end{aligned}\\
\begin{aligned}
s^{(1)}, n ;\,\widehat{\binom{1}{2}}{\overset{
{}_\curvearrowright}P_{\curlywedge}}E {\overset{
{}_\curvearrowleft}P_{\curlywedge}}-
H,\,\overline{\binom{1}{1}}SN: \text{ Example }
\Rightarrow &(c=0,\;d= 1/2,\; g=0,\; h=1)\\
& (\text{if }\mathcal{B}_1=0).
\end{aligned}
\end{gather*}

(c) {\it The case  $W_7=0$.} Since $\mu_2\ne0$ by
\eqref{val:S-e1-g0} we have $h\tau_1\tau_2=0$ and we consider two subcases:
$\widetilde M\ne0$ and $\widetilde M=0$.
\smallskip

(c.1) {\it The subcase  $\widetilde M\ne0$.} Then $h\ne0$ and we
have a node with coinciding eigenvalues.  By Remark
\ref{rem:CS-(0,0)} we may assume that this node is located at the
origin  of coordinates, i.e. we have $\tau_1=c^2 + 4 d=0$. So
we have a node $n^d$ (see Remark \ref{rem:S-e1-no n*}) and setting
$d=-c^2/4\ne0$ we may assume $c=1$ due to the
rescaling$(x,y,t)\mapsto(cx,y, t/c)$.  This leads to the family of
systems
\begin{equation} \label{CS:S-e1-g0a}
  \dot x =  x- y/4   + 2 h x y + y^2/4,\quad
 \dot y= x,
\end{equation}
and  we calculate
\begin{gather*}
\sigma=1 + 2 h y,\quad \mathcal{B}_1=-h (1 + 2 h)/2,\\
\mathcal{B}_2= -h^3 (1 + h),\quad \mathcal{F}_1=-h/2,\quad  \widetilde
M=-32 h^2 y^2.
\end{gather*}
Since $\widetilde M\ne0$  we could have a weak saddle if and only
if $h=-1/2$ and the weak saddle is of order one due to
$\mathcal{F}_1\ne0$ (see \cite{Vlp-NATMA}, Main Theorem, the
statement $(e_1)$). So considering  Lemma \ref{lem:ISP-mf=2} we
get the following two global configurations of singularities
\begin{gather*}
\begin{aligned}
s, n^d;\,\widehat{\binom{1}{2}}{\overset{
{}_\curvearrowright}P_{\curlywedge}}E {\overset{
{}_\curvearrowleft}P_{\curlywedge}}-
H,\,\overline{\binom{1}{1}} SN: \text{ Example }
\Rightarrow &(c=1,\; d= -1/4,\; g=0,\; h=1)\\
& (\text{if }\mathcal{B}_1\ne0);
\end{aligned} \\
\begin{aligned}
s^{(1)}, n^d ;\,\widehat{\binom{1}{2}}{\overset{
{}_\curvearrowright}P_{\curlywedge}}E {\overset{
{}_\curvearrowleft}P_{\curlywedge}}-
H,\,\overline{\binom{1}{1}}SN: \text{ Example }
\Rightarrow &(c=1,\; d= -1/4,\; g=0,\\
& h=-1/2)\quad (\text{if }\mathcal{B}_1=0).
\end{aligned}
\end{gather*}

(c.2) {\it The subcase  $\widetilde M=0$.} Then $h=0$ and we get
the family of systems
\begin{equation} \label{CS:S-e1-g0h0}
  \dot x =  cx+ dy  -d y^2,\quad
 \dot y= x,
\end{equation}
and  we calculate
 \begin{equation} \label{val:S-e1-g0h0}
\begin{gathered}
\widetilde M=0,\quad C_2=-d y^3,\quad \mu_2=d^2 y^2,\quad
 W_9=12(c^2 + 4 d) (c^2 - 4 d) =12\tau_1\tau_2,\\
\mathcal{T}_i=0,\quad i=1,2,3,4,\quad \sigma=c=\rho_1=\rho_2,\quad
\Delta_2=d=-\Delta_1.
\end{gathered}
\end{equation}
So the above systems possess a saddle and an anti-saddle and
clearly the type of the anti-saddle is governed by the invariant
polynomial $W_9$.
 \smallskip

($\alpha$) {\it The possibility $W_9<0$.} In this case we
have a saddle and a focus or a center and considering
\eqref{val:S-e1-g0h0} we observe that we could have a weak
singularity if and only if we have $\sigma=0$. However in this
case we get Hamiltonian systems possessing a center and an
integrable saddle.  As $C_2\ne0$ (due to $\mu_2\ne0$) considering
Lemma \ref{lem:ISP-mf=2} we obtain the following two  global
configurations of singularities:
\begin{gather*}
s, f;\,\widehat{\binom{2}{3}} {\overset{
{}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P- {\overset{
{}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P: \text{ Example } \Rightarrow (c=1,\;d= 1,\; g=0,\; h=0)
\quad (\text{if }\sigma\ne0);
\\
{\scriptstyle\$}, c;\,\widehat{\binom{2}{3}} {\overset{
{}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P- {\overset{
{}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P: \text{ Example } \Rightarrow
(c=0,\;d= 1,\; g=0,\; h=0)\quad (\text{if }\sigma=0).
\end{gather*}


($\beta$) {\it The possibility  $W_9\ge0$.} Then we have
a saddle and a node and in this case we have $\sigma\ne0$,
otherwise if $c=0$ we get $W_9=-192 d^2<0$. So the saddle is
strong and the node is generic if   $W_9>0$ and it is a node with
one direction (see Remark \ref{rem:S-e1-no n*}) if $W_9=0$.
Therefore considering Lemma \ref{lem:ISP-mf=2} we obtain the
following two  global configurations of singularities:
\begin{gather*}
s, n;\,\widehat{\binom{2}{3}} {\overset{
{}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P- {\overset{
{}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P: \text{ Example } \Rightarrow (c=3,\;d= 1,\; g=0,\; h=0)
\quad (\text{if }W_9>0);
\\
s, n^d;\,\widehat{\binom{2}{3}} {\overset{
{}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P- {\overset{
{}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P: \text{ Example } \Rightarrow (c=2,\;d= 1,\; g=0,\; h=0)
\quad (\text{if }W_9=0).
\end{gather*}
As all the possibilities are investigated we have ended  the
examination of systems \eqref{CS:S-e1}.


\subsection*{B. Systems \eqref{CS:S-e0}} We consider
systems \eqref{CS:S-e0}, i.e. the systems of the form
\begin{equation} \label{CS:S-ee0}
 \dot x =c x+d y -c x^2  + 2 h x y +k y^2,\quad
 \dot y= y,
\end{equation}
which possess the finite the singularities $M_1(0,0)$ and
$M_2(1,0)$. We observe that the Jacobian matrices corresponding to
these singular points  are respectively
$ \begin{pmatrix} c & d\\
0 &1\end{pmatrix}$ and $\begin{pmatrix}-c &  d+2h\\ 0 & 1\end{pmatrix}$ and
therefore we have
\begin{equation} \label{val:rho-i,Delta-i}
\rho_1=c+1, \quad \Delta_1=c, \quad \tau_1=(c-1)^2;\quad  \rho_2=1-c,
\quad \Delta_2=-c,\quad \tau_2=(c+1)^2.
\end{equation}
So obviously we have the next remark.

\begin{remark} \label{rem:S-e0-nd,n*} \rm
The family of systems \eqref{CS:S-ee0}   have a finite node and a finite
saddle. The node has coinciding eigenvalues  if and only if
$(c-1)(c+1)=0$ and in this case the  systems simultaneously have a
weak saddle. Moreover these systems  have a star node if and only if
either $c-1=d=0$, or $c+1=d+h=0$.
\end{remark}

For systems \eqref{CS:S-ee0}  we calculate
%
\begin{equation} \label{val:S-e0}
\begin{gathered}
\mu_0=\mu_1=\kappa=\widetilde K=0,\quad \eta=4 c^2 (h^2 + c k),\quad
\mu_2=c (c x^2 - 2 h x y - k y^2),\\
 \theta_2=0,\quad \mathbf{U}= c^2 y^2 (c x^2 - 2 h x y - k y^2)^2,
\quad G_8= -2 c^2 (h^2 + c k),\\
  W_4= 16  c^4 (c-1)^2(c+1)^2(h^2 + c k)^2,\\
\mathbf{D}=-192 c^4 (h^2 + c k), \quad \tilde L= 8 c (c x^2 - 2 h x y - k y^2),\\
\mathcal{F}_1=-2 c (c d - h + c h), \quad
\mathcal{T}_4=4 c^2 (c+1) (c-1) (h^2 + c k).\\
\end{gathered}
\end{equation}
We observe that for systems \eqref{CS:S-ee0} the following
conditions hold
\begin{equation} \label{cond:S-e0}
W_4\ge0,\quad \mu_2\tilde L>0, \quad \theta_2=0.
\end{equation}


\subsubsection*{The case $W_4\ne0$.} In this case by \eqref{cond:S-e0} we have
$W_4>0$ and this implies $\eta\mathcal{T}_4\ne0$. Hence  obviously
the node is generic and the saddle is strong.

Considering \eqref{cond:S-e0} and  Lemma \ref{lem:ISP-mf=2} we
obtain the following two global configurations  of singularities
\begin{gather*}
 s, n ;\,N^*,\,\binom{1}{1}\,\copyright,\binom{1}{1}\,
\copyright: \text{ Example } \Rightarrow (c=2,\;d=0,\; h=1,\;
k=-1) \;\; (\text{if }\eta<0);
\\
s, n ;\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}} NS,\,N^*: \text{ Example }
\Rightarrow (c=2,\;d=0,\; h=1,\; k=1) \quad (\text{if }\eta>0).
\end{gather*}

\subsubsection*{The case $W_4=0$.} Taking into account \eqref{val:S-e0}, we
consider two possibilities: $\eta\ne0$ and $\eta=0$.

\subsubsection*{The subcase $\eta\ne0$.} Then the condition
$W_4=0$ implies $\tau_1\tau_2=(c-1)^2(c+1)^2=0$, i.e. one of
singularities is a node with coinciding eigenvalues. According to
Remark \ref{rem:CS-(0,0)} we may assume that the singularity
$M_1(0,0)$ is such a node and this implies $ \tau_1 = (c-1)^2 =0$,
i.e. $c=1$. Then we calculate
\[
\mathcal{T}_4=\mathcal{F}_2=\mathcal{F}_3=0,\quad
\mathcal{T}_3\mathcal{F}=8 (h^2 + k)^2,\quad \mathcal{F}_1=-2d, \quad
\eta= 4 (h^2 + k).
\]
We observe that the condition $\eta\ne0$ implies
$\mathcal{T}_3\mathcal{F}>0$ and by \cite{Vlp-NATMA} we have a
weak saddle of order one if $\mathcal{F}_1\ne0$ and an
integrable saddle if  $\mathcal{F}_1=0$. On the other hand by
Remark \ref{rem:S-e0-nd,n*} we have a node $n^d$ if
$\mathcal{F}_1\ne0$ and a star node if $\mathcal{F}_1=0$.

Thus considering \eqref{cond:S-e0} and  Lemma \ref{lem:ISP-mf=2}
we obtain the following four global configurations  of
singularities:
\begin{gather*}
\begin{aligned}
s^{(1)}, n^d ;\,N^*,\,\binom{1}{1}\,\copyright,\binom{1}{1}\, \copyright:
\text{ Example }
\Rightarrow &(c=1,\;d=1,\; h=1,\; k=-2)\\
& (\text{if }\eta<0,\;
\mathcal{F}_1\ne0);
\end{aligned}\\
\begin{aligned}
{\scriptstyle\$}, n^* ;\,N^*,\,\binom{1}{1}\,\copyright,\binom{1}{1}\, \copyright:
\text{ Example } \Rightarrow &(c=1,\;d=0,\;h=1,\; k=-2) \\
&(\text{if }\eta<0, \; \mathcal{F}_1=0);
\end{aligned}\\
\begin{aligned}
s^{(1)}, n^d ;\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}} NS,\,N^*: \text{ Example }
\Rightarrow &(c=1,\; d=1,\; h=1,\; k=0) \\
&(\text{if }\eta>0,\; \mathcal{F}_1\ne0);
\end{aligned}\\
\begin{aligned}
 {\scriptstyle\$}, n^* ;\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}} NS,\,N^*: \text{ Example }
\Rightarrow &(c=1,\; d=0,\; h=1,\; k=0)  \\
&(\text{if }\eta>0,\; \mathcal{F}_1=0).
\end{aligned}
\end{gather*}


\subsubsection*{The subcase $\eta=0$.} Considering
\eqref{val:S-e0} due to $\mu_2\ne0$ (i.e. $c\ne0$) we obtain
$k=-h^2/c\ne0$  and this implies $W_4=0$. Then we may assume $h=1$
due to the rescaling $(x,y,t)\mapsto(x,y/h, t)$ . So we get the
family of systems
\begin{equation} \label{CS:S-e0-a}
  \dot x =  cx + dy - c x^2   + 2 x y - y^2/c,\quad
 \dot y= y,
\end{equation}
for which we calculate
\begin{equation} \label{val:S-e0-a}
\begin{gathered}
\kappa= \eta= \widetilde K=\mathbf{D}=\theta_5=0, \quad
 \kappa_1= -32 c (c-1+ cd)=16G_3,\\
 \widetilde M=-8 (c x - y)^2=-\tilde L,\quad \mu_2=(c x - y)^2,\quad
\sigma=1 + c - 2 c x + 2 y, \\
 W_8=2^{12}3^3 c^2 (c-1)^2 (1 + c)^2 (c-1+ c d)^2,\quad \mathcal{H}=0,\\
 \mathcal{T}_i=0,\quad i=1,2,3,4,\quad
 \mathcal{F}_1=-2(c-1+ cd),\quad \mathcal{B}_2= 4 c^2(c-1+ c d)^2,\\
 \mathcal{B}_1=2   c (c-1+ c d)(c+1)(c-1)=2   c (c-1+ c  d)\rho_1\rho_2.
\end{gathered}
\end{equation}
We note that by Remark \ref{rem:S-e0-nd,n*}  the above systems
possess   a node and a saddle.
\smallskip

(a) {\it The case $W_8\ne0$.}  Then by \eqref{val:S-e0-a} we have
$W_8>0$ and this implies $\kappa_1\mathcal{B}_1\ne0$. Hence
obviously the node is generic and the saddle is strong. Since
$\kappa_1\widetilde M\tilde L\ne0$ and $\theta_5=0$ considering
Lemma \ref{lem:ISP-mf=2} we obtain the configuration
\[
 s, n ;\,\widehat{\binom{2}{2}} {\overset{
  {}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P  {H_{ \curlywedge}}-H,\,N^*: \text{ Example }
 \Rightarrow (c=2,\; d=0 ,\; h= 1,\; k=-1/2).
\]

(b) {\it The case $W_8=0$.} Then $(c-1)(1 + c)(c-1+ c d)=0$ (as
$c\ne0$) and considering \eqref{val:S-e0-a} we have to distinguish
two subcases: $\kappa_1\ne0$ and $\kappa_1=0$.
\smallskip

(b.1) {\it The subcase $\kappa_1\ne0$.} Then $(c-1+ c d)\ne0$ and
hence we get $(c-1)(1 + c)=0$. So one of the singularities is a node
with coinciding eigenvalues and by Remark \ref{rem:CS-(0,0)} we
may assume that the singularity $M_1(0,0)$ is such a node, i.e.
$c=1$.  Then we calculate
\[
 \mathcal{H}=\mathcal{B}_1=0,\quad \mathcal{F}_1=-2d, \quad
 \mathcal{B}_2= 4 d^2,\quad \kappa_1=-32d.
\]
Therefore, the condition $\kappa_1\neq 0$ implies  $\mathcal{F}_1\ne0$
and $\mathcal{B}_2>0$. By
\cite{Vlp-NATMA} (see the Main Theorem, the statement $(e_1)$) and
by Remark~\ref{rem:S-e0-nd,n*}, systems \eqref{CS:S-e0-a} possess a
weak saddle of order one and a node $n^d$.

Thus considering the condition $\theta_5=0$ by Lemma
\ref{lem:ISP-mf=2} we get the  global configuration of
singularities
\[
s^{(1)}, n^d ;\,\widehat{\binom{2}{2}}
{\overset{
  {}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P  {H_{ \curlywedge}}-H,\,N^*: \text{ Example }
\Rightarrow (c=1,\;d=1 ,\; h= 1,\; k=-1).
\]

(b.2) {\it The subcase $\kappa_1=0$.} Then due to $\mu_2\ne0$
(i.e. $c\ne0$) by \eqref{val:S-e0-a} we have $d=(1-c)/c$. So for
systems \eqref{CS:S-e0-a} we calculate
\begin{equation} \label{val:S-e1-b0b}
\begin{gathered}
\widetilde K=\kappa_1=\theta_6=0,\quad \widetilde M=-8 (c x -
y)^2=\mathcal{G},\quad
 K_2= 96 c^2 (c x - y)^2, \\
\mu_2=(c x - y)^2,\quad \sigma=1 + c - 2 c x + 2 y,\quad
 \mathcal{F}_1=\mathcal{H}=\mathcal{B} =\mathcal{B}_1=\mathcal{B}_2=0,\\
\mathcal{B}_3= -72 (c-1) (1 + c) (c x - y)^2=-72\rho_1\rho_2(c x -y)^2.
\end{gathered}
\end{equation}
  Considering \eqref{val:S-e1-b0b} by \cite{Vlp-NATMA}
(see the Main Theorem, the statement $(e_3)[\delta])$ the saddle
will be weak (more precisely will be an integrable one) if and
only if $\mathcal{B}_3=0$. Moreover by Remark~\ref{rem:S-e0-nd,n*}
besides the integrable saddle we have a star node. Since $K_2>0$
and $\theta_6=0$   considering Lemma \ref{lem:ISP-mf=2} we obtain
the following  two global configurations of singularities:
\begin{gather*}
\begin{aligned}
s, n  ;\,\binom{2}{2}\,\overset{
{}_\curvearrowright}P\overset{
  {}_\curvearrowleft}PH-\overset{
  {}_\curvearrowright}P\overset{{}_\curvearrowleft}PH,\,N^*:
 \text{ Example } \Rightarrow  &(c=2,\; d=-1/2,\; h= 1,\; k=-1/2)\\
&(\text{if }\mathcal{B}_3\ne0);
\end{aligned}\\
\begin{aligned}
{\scriptstyle\$}, n^*;\,\binom{2}{2}\,\overset{
{}_\curvearrowright}P\overset{
  {}_\curvearrowleft}PH-\overset{
  {}_\curvearrowright}P\overset{{}_\curvearrowleft}PH,\,N^*:
\text{ Example } \Rightarrow &(c=1,\; d=0,\; h= 1,\; k=-1)\\
&(\text{if }\mathcal{B}_3=0).
\end{aligned}
\end{gather*}

As all   possible cases are examined, we have proved that the
family of systems with two distinct real finite singularities
possesses exactly  151 geometrically distinct global
configurations of singularities.


\subsection{The family of quadratic differential systems with only one
finite singularity which in addition is of multiplicity two}

 Assuming that
quadratic systems \eqref{sys:QSgen}   possess a double  singular
point, according to \cite{Vlp-NATMA} (see Table 2) we have to
consider two cases: $\widetilde K\ne0$ and  $\widetilde K=0$.

\subsubsection{Systems with $\widetilde K\ne0$.} In this case,
following \cite{Vlp-NATMA} (see Table 2), we consider the family
of systems
\begin{equation} \label{sys:16a}
\dot x=d y + g x^2 + 2 d x y,\quad
\dot y  = f y +  l x^2 + 2 f x y,
\end{equation}
possessing the double singular point  $M_{1,2}(0,0)$. For these
systems calculations yield
\begin{equation} \label{val:16a}
\mu_0=\mu_1=0,\quad \mu_2= (d l  - f g)^2 x^2,\quad
  \kappa= 128 d^2 (dl  - f g),\quad \mathcal{T}_4=4 d^2 f^2 (dl  - f g)^2.
\end{equation}

\begin{remark} \label{rem:16a-d=1,0} \rm
We observe that the family of systems \eqref{sys:16a} depends on
four parameters. However due to a rescaling we can reduce the
number of the parameters to two. More precisely since by the
condition  $\mu_2\ne0$  we have $d^2+f^2\ne0$, then we may
assume ${d,f}\in\{(1,1), (1,0), (0,1)\}$ due to the rescaling:
\begin{itemize}
\item[(i)] $(x,y,t)\mapsto(x, fy/d,t/f)$ if $df\ne0$;
\item[(ii)] $(x,y,t)\mapsto(x, y/d,t)$ if $f=0$ and
\item[(iii)] $(x,y,t)\mapsto(x, y,t/f)$ if $d=0$.
\end{itemize}
\end{remark}

Considering \eqref{val:16a} and $\mu_2\ne0$,   the condition $d=0$
is equivalent to $\kappa=0$ and in the case $\kappa\ne0$ the
condition $f=0$ is equivalent to $\mathcal{T}_4=0$.

\subsubsection*{The case $\kappa\ne0$.}

\subsubsection*{The subcase $\mathcal{T}_4\ne0$.} Then we have
$df\ne0$ and considering  Remark \ref{rem:16a-d=1,0} we may assume
$d=f=1$. So we obtain the 2-parameter family of systems
\begin{equation} \label{sys:16a)-d=f=1}
\dot x=  y + g x^2 + 2  x y,\quad
\dot y =   y +  l x^2 + 2  x y,
\end{equation}
for which calculations yield
% \label{sys:16a)-d=f=1}
\begin{gather*}
\mu_0=\mu_1=0,\quad \mu_2= (g-l)^2 x^2,\quad
 \widetilde K=  4 (g-l) x^2,\quad G_8=2(g-l)^2,\\
\mathcal{T}_4=4(g-l)^2=G_1,\quad
 \kappa= 128 (l - g), \quad
 \eta=4 \big[(g+2)^2 + 8 (l-g)\big],\\
 \widetilde M=-8 \big[(g-2)^2 + 6l\big] x^2 + 16 (2 - g) x y - 32 y^2.
\end{gather*}

\begin{remark} \label{rem:S-16a;kappa>0->eta>0} \rm
We observe that $ \widetilde M\ne0$ and  $\mu_2>0$. Moreover the
condition $\kappa>0$ implies $\eta>0$.
\end{remark}

As $G_8G_1\ne0$ according to \cite{Art-Llib-Vlp-IJBCh} (see Table
1, line 171) the double finite singular point is  a saddle-node.
\smallskip

(1) {\it The possibility $\kappa<0$.} As $\widetilde
M\ne0$ and  $\mu_2>0$, by Lemma \ref{lem:ISP-mf=2} we get the
following three global configurations of singularities:
\begin{gather*}
  \overline {sn}_{(2)};\, \overline{\binom{2}{1}}
N, \copyright,\copyright : \text{ Example } \Rightarrow (g=0,\; l=-1)
\quad (\text{if }\eta<0);
\\
\overline {sn}_{(2)};\, \overline{\binom{2}{1}}
N, S, N^\infty : \text{ Example } \Rightarrow (g=0,\;
l=-1/4) \quad (\text{if }\eta>0);
\\
\overline {sn}_{(2)};\, \overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}}N : \text{ Example }
\Rightarrow (g=0,\; l=-1/2) \quad (\text{if }\eta=0).
\end{gather*}

(2) {\it The possibility $\kappa>0$.}
 By Remark \ref{rem:S-16a;kappa>0->eta>0}
we have $\eta>0$ and   according to
 Lemma \ref{lem:ISP-mf=2}  we arrive at the
configuration
\[
\overline {sn}_{(2)};\, \overline{\binom{2}{1}} S, N^f, N^f:
\text{ Example } \Rightarrow (g=0,\; l=1).
\]

\subsubsection*{The subcase $\mathcal{T}_4=0$.} Then we have
$d\ne0$, $f=0$ and considering  Remark \ref{rem:16a-d=1,0} we may
assume $d=1$. So we obtain the 2-parameter family of systems:
\begin{equation} \label{sys:16a)-d=1,f=0}
 \dot x= y + g x^2 + 2 x y,\quad \dot y =  l x^2,
\end{equation}
for which calculations yield
\begin{equation} \label{val:16a)-d=1,f=0}
\begin{gathered}
\mu_0=\mu_1=0,\quad \mu_2=  l^2 x^2,\quad
 \widetilde K=  -4  l x^2,\quad G_8=2  l^2,\quad G_1=0,\\
 \kappa= 128 l, \quad
 \eta=4 (8l + g^2),\quad
 \widetilde M=-8 (6 cl + g^2) x^2 - 16 g x y - 32 y^2.
\end{gathered}
\end{equation}
As $G_8\ne0$ and $G_1=0$ according to \cite{Art-Llib-Vlp-IJBCh}
(see Table 1, line 175) the double finite singular point is  a
cusp.
\smallskip

(1) {\it The possibility $\kappa<0$.} As $\widetilde
M\ne0$ and  $\mu_2>0$, by
 Lemma \ref{lem:ISP-mf=2}  we get the following three global
configurations of singularities:
\begin{gather*}
\widehat{cp}_{(2)};\, \overline{\binom{2}{1}}
N, \copyright,\copyright : \text{ Example } \Rightarrow (g=2,\; l=-1)
\quad (\text{if }\eta<0);
\\
\widehat{cp}_{(2)};\, \overline{\binom{2}{1}}
N, S, N^\infty : \text{ Example } \Rightarrow (g=2,\;
l=-1/4) \quad (\text{if }\eta>0);
\\
\widehat{cp}_{(2)};\, \overline{\binom{0}{2}}
SN,\overline{\binom{2}{1}} N: \text{ Example }
\Rightarrow (g=2,\; l=-1/2) \quad (\text{if }\eta=0).
\end{gather*}

(2) {\it The possibility $\kappa>0$.} We observe that the
condition $\kappa>0$ implies $l>0$ and then $\eta=4 (8l + g^2)>0$.
So considering
 Lemma \ref{lem:ISP-mf=2}  we arrive at the global
configuration  of singularities
\[
\widehat{cp}_{(2)};\, \overline{\binom{2}{1}} S,
N^f, N^f : \text{ Example } \Rightarrow (g=1,\; l=1).
\]

\subsubsection*{The case $\kappa=0$.}
Then for systems  \eqref{sys:16a} we have $d=0$ and by Remark
\ref{rem:16a-d=1,0} we may assume $f=1$. So we get the family of
systems
\begin{equation} \label{sys:16a)-d=0,f=1}
\dot x =  g x^2,\quad \dot y = y+ l x^2 +2xy,
\end{equation}
for which calculations yield
\begin{equation} \label{val:16a)-d=0,f=1}
\begin{gathered}
\mu_0=\mu_1=0,\quad \mu_2= g^2 x^2,\quad
 \widetilde K=  4 g x^2,\quad \eta= \kappa= G_8=0,\quad K_2=0,\\
\tilde L=8 g(g-2) x^2,\quad \widetilde M=-8 (g-2)^2 x^2,\quad
C_2=- l x^3 + (g-2) x^2 y.
\end{gathered}
\end{equation}
As $G_8=0$ and $ \widetilde K\ne0$ according to
\cite{Art-Llib-Vlp-IJBCh} (see Table 1, line 172) the double
finite singular point is  a saddle-node.

\subsubsection*{The subcase $ \widetilde K<0$.} Then $g<0$ and
this implies  $ \widetilde M\ne0$.  As $\mu_2>0$ and $K_2=0$
considering  Lemma \ref{lem:ISP-mf=2}  we obtain the configuration
\[
\overline {sn}_{(2)};\, \binom{2}{2}\,\overset{
{}_\curvearrowright}PH-\overset{
  {}_\curvearrowright}PH,\,N^f: \text{ Example }
  \Rightarrow (g=-1,\ l=0).
\]

\subsubsection*{The subcase $ \widetilde K>0$.} In view of
Lemma \ref{lem:ISP-mf=2} we consider two possibilities: $\tilde
L\ne0$ and $\tilde L=0$.
\smallskip

(1) {\it The possibility $\tilde L\ne0$.} Then $g-2\ne0$
and we have $ \widetilde M\ne0$. So taking into account the
conditions $\mu_2>0$ and $K_2=0$, by  Lemma \ref{lem:ISP-mf=2} we
get the following two global configurations of singularities:
\begin{gather*}
\overline {sn}_{(2)};\, \binom{2}{2}\,\overset{
  {}_\curvearrowleft}PE-\overset{{}_\curvearrowleft}PE,S :
 \text{ Example }  \Rightarrow (g=1,\; l=0) \quad (\text{if }\tilde L<0);
\\
\overline {sn}_{(2)};\, \binom{2}{2}\overset{
{}_\curvearrowright}PH-\overset{
{}_\curvearrowright}PH,\,N^\infty: \text{ Example }
  \Rightarrow (g=3,\; l=0) \quad (\text{if }\tilde L>0).
\end{gather*}

(2) {\it The possibility $\tilde L=0$.} In this case $g=2$
and this implies $ \widetilde M=0$. As $\mu_2>0$ and $K_2=0$, by
 Lemma \ref{lem:ISP-mf=2}  we arrive at the following two
global configurations of singularities
\begin{gather*}
\overline {sn}_{(2)};\, \binom{2}{3} HE
-\overset{{}_\curvearrowleft}P\overset{
{}_\curvearrowleft}P:
 \text{ Example }  \Rightarrow (g=2,\; l=1) \quad (\text{if }C_2\ne0);
\\
\overline {sn}_{(2)};\, [\infty;\, N^d] : \text{ Example }
\Rightarrow (g=2,\ l=0) \quad (\text{if }C_2=0).
\end{gather*}

\subsubsection\textbf{Systems with $\widetilde K=0$.} In this case
according to \cite{Vlp-NATMA}  (see Table 2) we consider the
 family of systems
\begin{equation} \label{sys:16b}
\dot x=c x + d y,\quad \dot y  =   l x^2 + 2 m x y + n y^2,\quad
0\ne c^2 n - 2 c d m + d^2l  \equiv \mathcal{Z},
\end{equation}
for which we calculate
\begin{equation} \label{val:16b}
\begin{gathered}
\mu_0=\mu_1=\kappa=0,\quad \mu_2= \mathcal{Z}(l x^2 + 2 m x y + n
y^2),\quad  \widetilde K=0, \\
\tilde L=8 n (l x^2 + 2 m x y + n y^2),\quad
 \eta= 4 n^2 (m^2 -  l n),\\
  \theta_2=-d (m^2 -  l n),\quad
\widetilde M=-8 (4 m^2 - 3  l n) x^2 - 16 m n x y - 8 n^2 y^2,\\
  G_8=2 n (ln-m^2)\mathcal{Z},\quad G_1=4c^2n
  (ln-m^2)\mathcal{Z}=\mathcal{T}_4.
\end{gathered}
\end{equation}

\subsubsection*{The case $\tilde L\ne0$.} We consider two subcases:
$\eta\ne0$ and $\eta=0$.

\subsubsection*{The subcase $\eta\ne0$.} In this case $G_8\ne0$
and we   examine two possibilities: $\mathcal{T}_4\ne0$ and
$\mathcal{T}_4=0$.
\smallskip

(1) {\it The possibility $\mathcal{T}_4\ne0$.} Then we
have $cn\ne0$ and due to the rescaling $(x,y,t)\mapsto(x,
cy/n,t/c)$ we may assume $c=n=1$. So we get the family of systems
\begin{equation} \label{sys:16b)-n=c=1}
\dot x= x + d y,\quad \dot y  =   l x^2 + 2 m x y + y^2, \quad
0\ne1 - 2 d m + d^2l
\equiv \mathcal{Z}',
\end{equation}
where we may consider $d\in\{0,1\}$ due to the rescaling $x\to dx$
if $d\ne0$. For these systems we have
\begin{gather*}
\mathcal{T}_4=4  ( l - m^2)\mathcal{Z}'=G_1=2G_8,\quad \eta= 4
( m^2- l), \quad \theta_2=d ( l - m^2),\\
\mu_2=\mathcal{Z}'( l x^2 + 2 m x y + y^2),\quad \tilde L=8 (
l x^2 + 2 m x y + y^2).
\end{gather*}
As $G_8G_1\ne0$ according to \cite{Art-Llib-Vlp-IJBCh} (see Table
1, line 171) the   finite singular point is  a saddle-node.
\smallskip

(a) {\it  The case $\eta<0$.} Considering
 Lemma \ref{lem:ISP-mf=2}  we get the following two
global configurations of singularities:
\begin{gather*}
\overline {sn}_{(2)};\,N^d,\,\binom{1}{1}\copyright,\binom{1}{1}\copyright :
 \text{ Example }  \Rightarrow (d=1,\, l=1,\,m=0)\quad (\text{if }\theta_2\ne0);
\\
\overline {sn}_{(2)};\,N^*,\,\binom{1}{1}\,\copyright,\binom{1}{1}\, \copyright:
 \text{ Example }  \Rightarrow (d=0,\; l=1,\; m=0)\quad (\text{if }\theta_2=0).
\end{gather*}

(b) {\it The case $\eta>0$.} We observe, that
 $\operatorname{sign}(\mu_2\tilde L)=\operatorname{sign}(\mathcal{Z}')$.
Moreover if $d=0$ we obtain $\mathcal{Z}'=1>0$. So considering
 Lemma \ref{lem:ISP-mf=2}  we arrive at the following three
global configurations of singularities:
\begin{gather*}
\overline {sn}_{(2)};\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}} SN,\,N^d :
 \text{ Example }  \Rightarrow (d=1,\; l=0,\; m=1) \quad
(\text{if }\mu_2\tilde L<0);
\\
\begin{aligned}
\overline {sn}_{(2)};\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}} NS,\,N^d :
 \text{ Example }  \Rightarrow
&(d=1,\; l=0,\;m=-1)\\
&(\text{if }\mu_2\tilde L>0, \; \theta_2\ne0);
\end{aligned}\\
\begin{aligned}
\overline {sn}_{(2)};\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}} NS,\,N^* :
 \text{ Example }  \Rightarrow
&(d=0,\; l=0,\; m=1)\\
&(\text{if }\mu_2\tilde L>0,\; \theta_2=0).
\end{aligned}
\end{gather*}

(2) {\it The possibility $\mathcal{T}_4=0$.} By
\eqref{val:16b} due to the condition $\eta\ne0$ we obtain $c=0$ and
then $dln\ne0$. So via the   rescaling $(x,y,t)\mapsto(dx/n, y/n,t)$
we may assume $n=d=1$ and we arrive at  the family of systems
\begin{equation} \label{sys:16b)-n=1,c=0}
\dot x=    y,\quad \dot y  = l x^2 + 2 m x y + y^2.
\end{equation}
We observe that   we may assume $m\in\{0,1\}$ due to the rescaling
$(x,y,t)\mapsto(x, my ,t/m)$ if $m\ne0$. For these systems we
calculate
\begin{gather*}
G_1=0,\quad G_8=2 l (l - m^2),\quad \eta= 4
( m^2- l), \quad \theta_2=  l - m^2,\\
\mu_2=l( l x^2 + 2 m x y + y^2),\quad \tilde L=8 ( l x^2 + 2 m x y + y^2).
\end{gather*}
So the condition $\eta\mu_2\ne0$ implies $G_8\ne0$ and due to
$G_1=0$  by \cite{Art-Llib-Vlp-IJBCh} (see Table 1, line 175) the
double finite singular point is  a cusp.

On the other hand we have $\theta_2\ne0$ and $\operatorname{sign}(\mu_2\tilde
L)=\operatorname{sign}(l)$. So
  considering  Lemma \ref{lem:ISP-mf=2}  we get the
following three global configurations of singularities:
\begin{gather*}
\widehat{cp}_{(2)};\,N^d,\,\binom{1}{1}\copyright,\binom{1}{1} \copyright:
 \text{ Example }  \Rightarrow (l=1,\; m=0) \quad (\text{if }\eta<0);
\\
\widehat{cp}_{(2)};\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}} SN,\,N^d :
 \text{ Example }  \Rightarrow (l=-1,\; m=1) \quad
 (\text{if }\eta>0,\; \mu_2\tilde L<0);
\\
\widehat{cp}_{(2)};\,\overline{\binom{1}{1}}
SN,\,\overline{\binom{1}{1}} NS,\,N^d :
 \text{ Example }  \Rightarrow (l=1,\;m=2) \quad
 (\text{if }\eta>0,\; \mu_2\tilde L>0).
\end{gather*}

\subsubsection*{The subcase $\eta=0$.}
As $n\ne0$ (due to $\tilde L\ne0$) we may assume $n=1$ due to a rescaling.
So considering \eqref{val:16b} the condition $\eta=0$ gives $m^2-l=0$  and  we
obtain $l=m^2$. Then for systems \eqref{sys:16b} we have
$$
\mu_2=(c - d m)^2 (m x + y)^2,\quad \kappa_1=32 m (c - d m),
$$
and as $\mu_2\ne0$ the condition $\kappa_1=0$ is equivalent to
$m=0$.
\smallskip

(1) {\it The possibility  $\kappa_1\ne0$.} Then $m\ne0$ and
we may assume $m=1$ due to the rescaling  $x\to x/m$. Therefore  we
arrive at  the family of systems
\begin{equation} \label{sys:16b)-n=1,l=m2}
\dot x=  cx+ d y,\quad \dot y  = (x+y)^2,
\end{equation}
for which  we calculate
\begin{equation} \label{val:16b)-n=1,l=m2}
\begin{gathered}
 \eta=\kappa= G_8=0,\quad \mu_2= (c - d)^2 (x + y)^2,\quad
 \widetilde K=\widetilde N=0, \\
\tilde L=8 (x + y)^2,\quad
  \widetilde M=-8 (x + y)^2,\quad \theta_5= 96 (c - d) d (x + y)^3,\\
\kappa_1= 32 (c - d),\quad   F_3=24 c (c - d) (x + y),\quad
\mathcal{B}_1=2 c^2 (c - d)^2.
\end{gathered}
\end{equation}

(a) {\it The case $\mathcal{B}_1\ne0$.} Then $c\ne0$ and we  may
assume $c=1$ due to the rescaling $(x,y,t)\mapsto(cx, cy ,t/c)$.
We observe that in this case $F_3\ne0$ and as $G_8=\widetilde K
=\widetilde N=0$, according to \cite{Art-Llib-Vlp-IJBCh} (see
Table 1, line 174), the   finite singular point is  a saddle-node.

On the other hand we have   $\eta=\kappa=\widetilde K=0$ and
$\widetilde M\tilde L\kappa_1\ne0$. Therefore considering
 Lemma \ref{lem:ISP-mf=2}  we obtain the following two
global configurations of singularities:
\begin{gather*}
\overline {sn}_{(2)};\, \widehat{\binom{2}{2}}
{\overset{   {}_\curvearrowright}P_{\curlywedge}}\overset{
  {}_\curvearrowleft}P  {H_{ \curlywedge}}-H,\,N^d: \text{ Example }
  \Rightarrow (c=1,\,d= 2) \quad (\text{if }\theta_5\ne0);
\\
\overline {sn}_{(2)};\, \widehat{\binom{2}{2}}
{\overset{   {}_\curvearrowright}P_{\curlywedge}}\overset{
  {}_\curvearrowleft}P  {H_{ \curlywedge}}-H,\,N^* : \text{ Example }
 \Rightarrow (c=1,\; d= 0) \quad (\text{if }\theta_5=0).
\end{gather*}

(b) {\it The case $\mathcal{B}_1=0$.} Then $c=0$   and this
implies $F_3=0$. So   by \cite{Art-Llib-Vlp-IJBCh} (see Table 1,
line 177) the double finite singular point is  a cusp. As in this
case $\theta_5=-96 d^2 (x + y)^3\ne0$ (due to $\mu_2=d^2 (x +
y)^2\ne0$), considering \eqref{val:16b)-n=1,l=m2} and Lemma
\ref{lem:ISP-mf=2}  we get the configuration
\[
\widehat{cp}_{(2)};\,\widehat{\binom{2}{2}} {\overset{
{}_\curvearrowright}P_{\curlywedge}}\overset{
  {}_\curvearrowleft}P  {H_{ \curlywedge}}-H,\,N^d:
 \text{ Example }  \Rightarrow (c=0,\;d=1).
\]


(1) {\it The possibility $\kappa_1=0$.} In this case we
have $m=0$ and this leads to the family of systems
\begin{equation} \label{sys:16b)-n=1,l=m=0}
\dot x=  cx+ d y,\quad \dot y  = y^2,
\end{equation}
for which  we calculate
\[
\mu_2=c^2 y^2,\quad F_3=24 c^2 y,\quad K_2=0,\quad \theta_6=-8 d y^4
\]
and therefore the condition
$\mu_2\ne0$ implies $F_3\ne0$. So the double finite singularity is
a saddle-node and   considering
 Lemma \ref{lem:ISP-mf=2}  we obtain the following two
global configurations of singularities:
\begin{gather*}
\overline {sn}_{(2)};\, \binom{2}{2}\,\overset{
{}_\curvearrowright}PH-\overset{
  {}_\curvearrowright}PH,\,N^d: \text{ Example }
\Rightarrow (c=1,\;d= 1)\quad (\text{if }\theta_6\ne0);
\\
\overline {sn}_{(2)};\, \binom{2}{2} \overset{
{}_\curvearrowright}PH-\overset{
  {}_\curvearrowright}PH,\,N^*: \text{ Example }
\Rightarrow (c=1,\; d= 0) \quad (\text{if }\theta_6=0).
\end{gather*}

\subsubsection*{The case $\tilde L=0$.} Then for systems
\eqref{sys:16b} we have $n=0$ and then $d\ne0$ (otherwise we get
degenerate systems). So we may assume $d=1$ (due to a rescaling)
and we obtain the family of systems
\begin{equation} \label{sys:16b)-n=0}
\dot x=c x +  y,\quad  \dot y =   l x^2 + 2 m x y,
\end{equation}
for which we calculate
\begin{equation} \label{val:16b)-n=0}
\begin{gathered}
 \eta=\kappa= G_8=\widetilde K=\tilde L= 0,\quad
\mu_2= (l - 2 c m) x (l x + 2 m y),\quad
 \widetilde N=-4 m^2 x^2, \\
\widetilde M=-32 m^2 x^2,\quad
  G_{10}= c^2 m^3 (l - 2 c m),\quad \mathcal{B}_1=2c^2 m  (l - 2 c m).
\end{gathered}
\end{equation}

\subsubsection*{The subcase $\widetilde M\ne0$.}
Then $m\ne0$ and
we may assume $m=1$ due to the rescaling $(x,y,t)\mapsto( x/m,
y/m,t)$. Moreover for systems above with $m=1$ we may consider
$c\in\{0,1\}$ due to the rescaling $(x,y,t)\mapsto(cx,c^2 y,t/c)$ if
$c\ne0$.

Therefore we have $\widetilde N\ne0$ and by
\cite{Art-Llib-Vlp-IJBCh} (see Table 1, lines 173,176) the finite
singular point is a saddle--node if $G_{10}\ne0$ and it is a cusp
if $G_{10}=0$. We observe that the condition $G_{10}=0$ is
equivalent to $\mathcal{B}_1=0$. So as $\widetilde M\ne0$ and
$\tilde L= 0$, considering  Lemma \ref{lem:ISP-mf=2}  we obtain
the following two global configurations of singularities:
\begin{gather*}
\begin{aligned}
\overline {sn}_{(2)};\, \widehat{\binom{1}{2}}
{\overset{ {}_\curvearrowright}P_{\curlywedge}}E
{\overset{ {}_\curvearrowleft}P_{\curlywedge}}-
H,\,\overline{\binom{1}{1}} SN : \text{ Example }
 \Rightarrow &(c= 1,\; m= 1,\; l=1)\\
& (\text{if }\mathcal{B}_1\ne0);
\end{aligned}\\
\begin{aligned}
\widehat{cp}_{(2)};\, \widehat{\binom{1}{2}}
{\overset{ {}_\curvearrowright}P_{\curlywedge}}E
{\overset{ {}_\curvearrowleft}P_{\curlywedge}}-
H,\,\overline{\binom{1}{1}}SN : \text{ Example }
  \Rightarrow &(c= 0,\; m= 1,\; l=1)\\
(\text{if }\mathcal{B}_1=0).
\end{aligned}
\end{gather*}

\subsubsection*{The subcase  $\widetilde M=0$.} Then $m=0$ and
$l\ne0$ (otherwise we get degenerate systems). Hence we may assume
$l=1$ due to the rescaling $(x,y,t)\mapsto( x/l, y/l,t)$. Then for
systems \eqref{sys:16b)-n=0} with $m=0$ and $l=1$ considering
\eqref{val:16b)-n=0} we obtain
$$
\widetilde M=\widetilde K=\widetilde N=G_8=0,\quad C_2=-x^3,\quad
F_3=-24cx,\quad \mathcal{B}_4=6 c x^2 (c x + y).
$$
So by \cite{Art-Llib-Vlp-IJBCh} (see Table 1, lines 174,177) the
finite singular point is a saddle--node if $F_3\ne0$ and it is a
cusp if $F_3=0$. We observe that the condition $F_3=0$  is
equivalent to $\mathcal{B}_4=0$. Considering the conditions above
according to  Lemma \ref{lem:ISP-mf=2}  we obtain the following
two global configurations of singularities
\begin{gather*}
\overline {sn}_{(2)};\, \widehat{\binom{2}{3}}
{\overset{{}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P- {\overset{
{}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P  : \text{ Example }
 \Rightarrow (c= 1,\; m= 0,\; l=1) \quad (\text{if }\mathcal{B}_4\ne0);
\\
\widehat{cp}_{(2)};\, \widehat{\binom{2}{3}}
{\overset{{}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P- {\overset{
{}_\curvearrowright}P_{\curlywedge}}\overset{
{}_\curvearrowleft}P : \text{ Example }  \Rightarrow (c= 0,\;m= 0,\;
l=1)\quad (\text{if }\mathcal{B}_4=0).
\end{gather*}
Since  all   possibilities   are examined for this case, we have
proved that the family of systems with a single finite real
singular point which is of multiplicity two  possesses exactly 30
geometrically distinct global configurations of singularities.

With this the whole proof of our Main Theorem is complete.


\subsection*{Acknowledgments}
The authors are grateful to the anonymous referee for his/her comments
and suggestions.

The first two authors are partially supported by a MINECO/FEDER
grant number MTM2008-03437
and an AGAUR grant number 2009SGR 410.
The second author is also supported by ICREA Academia and
FP7-PEOPLE2012-IRSES number 318999.
The second and fourth authors are partially supported by
the grant FP7-PEOPLE-2012-IRSES-316338.
The third author is supported by NSERC-RGPIN (8528-2010).
The fourth author is also supported by grant 12.839.08.05F from
 SCSTD of ASM and partially by NSERC-RGPIN (8528-2010).

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\end{document}