\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{multirow}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 64, pp. 1--69.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/64\hfil Global topological classification]
{Global topological classification of Lotka-Volterra quadratic
 differential systems}

\author[D. Schlomiuk, N. Vulpe \hfil EJDE-2012/64\hfilneg]
{Dana Schlomiuk, Nicolae Vulpe}

\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}
\email{nvulpe@gmail.com}

\thanks{Submitted January 19, 2012. Published April 25, 2012.}
\thanks{Research supported by NSERC }
\subjclass[2000]{58K30, 34A26,34C05, 34C40}
\keywords{Quadratic vector fields; Lotka-Volterra  differential systems;
\hfill\break\indent
 phase portraits; affine invariant polynomials; topological invariants}

\begin{abstract}
 The Lotka-Volterra planar quadratic differential systems have
 numerous applications but the global study of this class proved to
 be a  challenge difficult to handle.  Indeed, the four attempts to
 classify them (Reyn (1987), W\"orz-Buserkros (1993),
 Georgescu (2007) and  Cao and Jiang (2008)) produced results which
 are not in agreement.  The lack of adequate global classification
 tools for  the large number of phase portraits encountered,
 explains this situation. All Lotka-Volterra systems possess
 invariant straight lines, each with its own multiplicity. In this
 article we use as a   global classification tool for
 Lotka-Volterra systems the concept of configuration of invariant
 lines (including the line at infinity). The class splits according
 to the types of configurations in smaller subclasses which makes
 it easier to have a good control over the phase portraits in each
 subclass.  At the same time the classification becomes more
 transparent and easier to grasp.  We obtain a total of 112
 topologically distinct phase portraits: 60 of them with exactly
 three invariant lines, all simple; 27 portraits with invariant
 lines with total multiplicity at least four; 5 with the line at
 infinity filled up with singularities; 20 phase portraits of
 degenerate systems.  We also make a thorough analysis of the
 results in the paper of Cao and Jiang \cite{CaoJia08}. In contrast
 to the results on the classification in \cite{CaoJia08}, done in
 terms of inequalities on the coefficients of normal forms, we
 construct   invariant criteria for distinguishing these portraits
 in the whole parameter space $\mathbb{R}^{12}$ of coefficients.
\end{abstract}

\maketitle

\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{observation}[theorem]{Observation}
\newtheorem{notation}[theorem]{Notation}
\allowdisplaybreaks
\renewcommand*{\tablename}{Diagram}
\tableofcontents

\section{Introduction}

  In this article we consider real autonomous  differential systems
\begin{equation}\label{sys:PSgen} %(S)
 \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 $\mathbb{R}$ and  their associated vector fields
\begin{equation}\label{vect:field}
     \tilde D=p(x,y)\frac{\partial}{\partial x} + q(x,y)\frac{\partial}{\partial
     y}.
\end{equation}
We call \emph{degree} of a system \eqref{sys:PSgen} (or of a vector field
\eqref{vect:field}) the integer
$$ n=\max(\deg p,  \deg q).
$$
In particular we call \emph{quadratic} a differential system
\eqref{sys:PSgen} with $n=2$ and we denote by QS the
class of all such systems.

The study of quadratic differential systems is motivated in part
by their many applications.  On the other hand they are also
interesting for theoretical reasons. Indeed, hard problems on
polynomial differential systems, among them Hilbert's 16th
problem, have been open for more than a century even for the
quadratic case. These problems are of a global nature and while
the global study of the whole quadratic class is not within reach
at this time, a handful of specific subfamilies of this class have
been successfully studied globally.

The goal of this article is to give a complete global topological
classification of the subfamily of quadratic differential systems
which can be brought by affine transformation of the form
\begin{equation}\label{sys:gen-LV}
\begin{gathered}
\dot{x}=x(a_0+a_1x+a_2y)\equiv p(x,y), \\
\dot{y}=y(b_0+b_1x+b_2y)\equiv q(x,y),
\end{gathered}
\end{equation}
where $p$, $q$ are polynomials in $x,y$ with real coefficients and
$\max(\deg(p), \deg(q))=2$.

Usually we say that two systems $(S_1)$ and $(S_2)$ are
topological equivalent if and only if there exists a homeomorphism
of the plane carrying orbits to orbits and preserving their
orientations. In this paper we say that $(S_1)$ and $(S_2)$ are
\emph{topological equivalent} if and only if there exists a
homeomorphism of the plane carrying orbits to orbits and
preserving or reversing their orientations. We use this definition
in order to halve the number of phase portraits.

 Systems \eqref{sys:gen-LV} are
called Lotka-Volterra as they were proposed independently by
Alfred J. Lotka in 1925 \cite{Lot25} and Vito Volterra in 1926
\cite{Vol26}. Actually  Lotka and Volterra considered  initially
the systems \eqref{sys:gen-LV} with $a_1=0=b_2$ but in the current
literature are called Lotka-Volterra, systems of the more general
form \eqref{sys:gen-LV}. The scientific literature on this family
has been steadily growing due to their many applications (see
\cite{SchVul10-JFPTA} for references on applications).

It is   estimated that the class of quadratic differential systems
will yield more than two thousand topologically distinct phase
portraits. The study of subfamilies of the quadratic class and in
particular the Lotka-Volterra family, forms a good testing ground
for the analogous but much more difficult task of classifying the
whole quadratic class.

The global study of several families of quadratic vector fields
was completely done. Examples of such  families are:

\begin{itemize}

\item the  quadratic vector fields possessing a center \cite{Vulpe-DU}, \cite{Schlomiuk:1993}
 \cite{Zoladek:1994}, \cite{Pal-Schlomiuk:1997};
\item the   quadratic Hamiltonian vector fields \cite{Art-Llib}, \cite{ArtLliVul-EJDE};
\item the  quadratic vector fields with invariant
straight lines of total multiplicity at least four
\cite{SchVul08-RMJM}, \cite{SchVul08-BASM};
\item the   planar quadratic differential systems
possessing a line of singularities at infinity
\cite{SchVul08-JDDE};
\item the   quadratic vector fields possessing an integrable saddle
\cite{ArtLliVul-Prep2011}.

\end{itemize}
All the systems in the above mentioned classes are integrable.
Indeed, every quadratic Hamiltonian system has a cubic polynomial
as first integral. All the  systems occurring in the other
families above are proven to be integrable on the complement of an
algebraic curve in the papers mention above,  via the
algebro-geometric method of Darboux. The global study of each of
the above classes was done using only algebraic methods. This is
essentially due to the existence of Darboux   \emph{inverse
integrating factors} (i.e. $1/R(x,y)$, where $ R(x,y)$ is an
integrating factor).


While the global study of QS is a distant goal at this time, the
global study of the infinite singular points of systems in QS was
done. Furthermore this was achieved by using only algebraic and
geometric  methods  in \cite{NikVul}, \cite{SchVul05-JDE}.  The
global study of the finite singularities was also done by using
only algebraic and algebro-geometrical methods in
\cite{ArtLliVul08-IJBCh}.

 In \cite{Cop}, Coppel wrote:

\begin{quote}
Ideally one might hope to characterize the phase
portraits of quadratic systems by means of algebraic inequalities
on the coefficients.  However, attempts in this direction have met
with very limited success \dots
\end{quote}
This task proved to be impossible. Indeed, Dumortier and
Fiddelaers \cite{Dum-Fid} and Roussarie \cite{Roussarie} exhibited
examples of families of quadratic vector fields which have
non-algebraic bifurcation sets. The following two  classes of
quadratic vector fields were studied globally by algebraic methods
coupled with analytic and numerical methods:

\begin{itemize}
\item the family of quadratic vector fields with a weak focus of third order \cite{Llibre-Schlomiuk:2004};
\item the family of quadratic vector fields with a weak focus of second order
\cite{Art-Llib-Schl}.
\end{itemize}
We point out that in both families, besides algebraic
hyper-surfaces of bifurcation points,  there are non-algebraic
hyper-surfaces of bifurcation points in the parameter space.


It is natural to ask the following questions:
\begin{quote}
How much of the behavior of quadratic (or more generally
polynomial) vector fields or  how far can we go in their global
theory  by using mainly algebraic means?
\end{quote}


Modulo the action of the group of affine transformations and time
homotheties, the planar Lotka-Volterra class is 3-dimensional
while the class of quadratic differential systems modulo the same
group action is 5-dimensional. Due to the global result saying
that any system in the Lotka-Volterra class has no limit cycles
(see Theorem \ref{thm:Bautin} or \cite{Bautin}, \cite{Cop}), it is
possible to draw the bifurcation diagram of this class.


 The literature on  the
Lotka-Volterra equations has become quite ample. In particular
there were several attempts to give complete classifications of
this family \cite{Rey87}(1987), \cite{Wor-Bus93} (1993),
\cite{Geo07} (2007), \cite{CaoJia08} (2008). A quick check of the
references given in the last three papers indicates that none of
these authors mentions anyone of the previously published papers
suggesting that they were not aware of them.  But did they obtain
results which are in agreement? In fact they are not.  This was
shown for the first three articles above in \cite{SchNai08}.  In
this work we shall also discuss the results in \cite{CaoJia08} and
we show that our results are not everywhere in agreement with the
results in \cite{CaoJia08}. Indeed, we prove here that several
phase portraits   claimed   to be topologically distinct in
\cite{CaoJia08}  are in fact topologically equivalent, and we
indicate 8 phase portraits which are incorrect.  On the other hand
we point out the following observation:

\begin{observation}\label{obs:dif-LVs} \rm
While in this article a system \eqref{sys:gen-LV} is taken to have\\
$\max(\deg(p), \deg(q))=2$, in \cite{CaoJia08} the authors only
consider systems \eqref{sys:gen-LV} with $ \deg(p)=\deg(q)=2$.
Naturally they got fewer phase portraits than we get here. More
precisely 8 phase portraits in our classification do not appear in
\cite{CaoJia08} and we indicate them by writing  the word
``\emph{omitted}" on the lines where they appear  in  the
diagrams (see Diagram \ref{Fig:N=5} --
Diagram \ref{Fig:N=inf}).
\end{observation}

To obtain a clear, transparent classification, one needs
to have powerful  global classifying tools and they are missing in
\cite{CaoJia08}. In our view, the main shortcoming is neither the
fact that some portraits are incorrect or missing, nor that in the
list we have topologically equivalent phase portraits which are
claimed to be distinct.  The main shortcoming of this
classification is that it is not helpful for understanding
globally this class. Indeed, the classification is done in terms
of inequalities on the coefficients of several normal forms for
the systems. Since there are  so many phase portraits we end up
with pages of inequalities  giving us no insights into the global
phenomena present for this class.

In contrast in this article we use powerful classification tools
such as  \emph{affine and topological invariants}. The
Lotka-Volterra equations have an inherent algebro-geometric
structure.  We spelled out this algebro-geometric structure in
\cite{SchVul10-JFPTA}.  We used the number of distinct invariant
straight lines, as well as their multiplicities, as a basic global
geometric classifying tool. We combine this algebro-geometric
information with information involving the real singularities of
the systems located on these invariant lines in the concept of
\emph{configuration of invariant straight lines of a system},
introduced in \cite{SchVul04-QTDS} (see Section
\ref{sec:Global...} further below). This is an example of an
affine invariant.


The topological equivalence relation is distinct from the affine
equivalence relation and it is in fact coarser than the affine
one. However the affine equivalence relation is a very powerful
tool for computation and it is of great use in the topological
classification due to the possibility of using   an arsenal  of
specific affine invariant polynomials. We base the topological
classification on the affine classification of the
 configurations of invariant lines of Lotka-Volterra
systems obtained by us in \cite{SchVul10-JFPTA}. Our results split
this class into subclasses according to the possibilities we have
for the types of configurations occurring for this class. We then
focus our attention on these subclasses, each of which has much
fewer phase portraits and thus it is much easier to keep track of
them. By using this approach we are  also able to give necessary
and sufficient conditions in terms of polynomial invariants for
the realization of each one of the phase portraits.

 Clearly, any system which could be brought by affine
transformation and time homotheties to a system \eqref{sys:gen-LV}
has the same geometric properties as \eqref{sys:gen-LV}.

\begin{definition} \rm
We denote by  LV the class of all planar differential
systems which  could be brought by affine transformations and time
rescaling to the form \eqref{sys:gen-LV} above and the systems in
this class will be called LV-systems.
\end{definition}


In \cite{SchVul10-JFPTA}, it was shown that the LV quadratic
systems form a subset of an algebraic set in the   parameter space
$\mathbb{R}^{12}$ of coefficients.  The goal of our work here is to give a
rigorous and complete \emph{topological} classification of the
phase portraits of this class and to construct   invariant
criteria for distinguishing these portraits.


Our Main Theorem is the following:

\begin{theorem} \label{mainthm}
 The class of all
Lotka-Volterra quadratic differential systems has a total of 112
topologically distinct phase portraits.  Among these, 60 portraits
are for systems with three simple invariant lines; 27 are
portraits of systems with invariant lines of total multiplicity at
least four; 5  phase portraits are for Lotka-Volterra systems
which have the line at infinity filled up with singularities;  20
phase portraits are for the degenerate systems.

(i) Consider the 13 configurations Config. 3.j,
$j\in\{1,\ldots,13\}$ (see Definition \ref{def2.2}) 
with three simple invariant lines given in
Fig. \ref{Fig:Config-FinNs}. For each configuration
{Config. 3.j} we have a number $n_j$ of topologically
distinct phase portraits.   Then $\sum_{j=1}^{13} n_j=65 $ and the
65 phase portraits (not necessarily topologically distinct) are
given in Fig. \ref{Fig:Ph-Port-3ILs}. The necessary and
sufficient affine invariant conditions for the realization of each
one of these portraits are given in Table 5.

(ii) Consider the 34 configurations of Lotka-Volterra
systems {Config. 4.1},\ldots, {Config. 6.8} with
invariant lines of total multiplicity  at least four given in
Fig. \ref{Fig:Config-FinNs}. For each one of these 34
configurations we have a number   $m_i$, $i\in\{1,\ldots,34\}$  of
topologically distinct phase portraits. Then $\sum_{i=1}^{34}
m_i=59 $ and the 59 phase portraits (not necessarily topologically
distinct) are given in Fig. \ref{Fig:Ph-Port:4-6IL}. The
necessary and sufficient affine invariant conditions for the
realization of each one of these portraits are given in Table 3.


(iii)  Consider the 4 configurations of Lotka-Volterra
systems {Config. $C_2.$j}, $j\in\{1,2,5,7\}$  with the line
at infinity filled up with singularities given in Fig.
\ref{Fig:Config-InfNs}. For each one of these 4 configurations we
have a unique   phase portrait, except for the configuration
{Config. $C_2.$j} for which we have two phase portraits.
The 5 phase  portraits are   topologically distinct and they  are
given in Fig. \ref{Fig:Pictures:C2-0}. The necessary and
sufficient affine invariant conditions for the realization of each
one of these portraits are given in Table 3.


(iv) Consider the 14 configurations
{Config. LV$_d$.j}, $j\in\{1,\ldots,14\}$   given in
Fig. \ref{Fig:Config-InfNs}, of  the degenerate
quadratic Lotka-Volterra systems. For each configuration
Config. LV$_d$.j we have a number $s_j$ of topologically
distinct phase portraits. Then $\sum_{j=1}^{14} s_j=20 $ and the
20  phase portraits   given in
Fig. \ref{Fig:Phase-Port-Deg} are topologically
distinct. The necessary and sufficient affine invariant conditions
for the realization of each one of these portraits are given in
Table 6.

(v) Of the 149 phase portraits obtained by listing those
occurring in the classes (i)--(iv), only 112 are
topologically distinct (see  Diagrams  \ref{Fig:N=7}--\ref{Fig:N=inf}).
\end{theorem}


To characterize each phase portrait  we use   affine invariant
polynomials  which  we define in Subsection~\ref{subs: pol-inv};
to prove that two arbitrarily chosen phase portraits in  the
complete list are topologically distinct we use topological
invariants which  we define in
Subsection~\ref{sec:topol-distinct}.

\begin{observation} \rm
In Fig. \ref{Fig:Ph-Port-3ILs} we see the phase portraits
of the family of LV-systems with exactly three invariant lines.
In this figure we have 23 couples of phase portraits such that in
any such couple the phase portraits are topologically equivalent
and they are distinguished in the picture by the presence of a
focus instead of a node (see for example, {Picture 3.1(a2)}
and {Picture 3.1($\stackrel{\,*}{a}$2)}). Our algebraic
apparatus allows us to distinguish within each couple the two
phase portraits by  algebraic means.
\end{observation}

 The article is organized as follows: In Section
\ref{sec:Global...} we give the definitions of the global concepts
used in this article, such as for example the notion of
\emph{configuration of invariant lines} and we  state the
theorem proved in \cite{SchVul10-JFPTA} classifying the
Lotka-Volterra differential systems according to their
configurations of invariant lines.  We  also state    results
which we need and which were obtained in \cite{SchVul04-QTDS},
\cite{SchVul08-RMJM}, \cite{SchVul08-NATMA}, \cite{SchVul08-BASM},
\cite{SchVul08-JDDE},\cite{SchVul10-JFPTA}. In Section
\ref{sec:proof-MT} we prove the Main Theorem and we give some
concluding comments.


\section{Global geometric concepts and preliminary
results}\label{sec:Global...}

Our classification is based on the concept of
\emph{configuration of invariant lines} of a differential system
and on results obtained in \cite{SchVul10-JFPTA}.

The concept of invariant algebraic curve of a differential system
is due to Darboux \cite{Dar1878}. Roughly speaking these are
algebraic curves which are unions of phase curves. The presence of
such algebraic invariant curves is an important information about
a system. For example if we have sufficiently many such curves,
the system is integrable, i.e. it has a non-constant analytic
first integral on the complement of some algebraic curve
(\cite{Dar1878}). The following is the formal definition due to
Darboux.

\begin{definition} {\rm
 An \emph{affine algebraic invariant  curve} (or an \emph{algebraic
particular integral}) of a polynomial system \eqref{sys:PSgen} or
of a vector field \eqref{vect:field} is a curve $f(x,y)=0$ where
$f\in \mathbb{C}[x,y]$, $\deg(f)\ge1$, such that there exists $k(x,y)\in
\mathbb{C}[x,y]$ satisfying $\tilde Df=fk$ in $\mathbb{C}[x,y]$. We call $k$ the
\emph{cofactor} of $f(x,y)$ with respect to the system.}
\end{definition}

We stress the fact that   we have $f(x,y)\in \mathbb{C}[x,y]$. This is
important because even in the case when we are only interested in
integrability of real systems, the complex invariant curves are
helpful in the search for a real first integral of the systems.

If a planar polynomial differential system has invariant algebraic
curves then these curves could have \emph{multiplicities}. Just
as a singularity of a system could be a multiple singularity,
meaning that in perturbations this singularity splits into two or
more singularities, so also algebraic invariant curves could have
multiplicities, meaning that in neighboring systems this curve
splits into two or more invariant algebraic curves. In
\cite{ChrLliPer2007}  the authors define several notions of
multiplicity of invariant curves and show that they coincide for
irreducible invariant curves under some "generic" conditions.

In this work we shall only need invariant straight lines and their
multiplicities (we work with the definitions given in
\cite{SchVul04-QTDS}).  All planar Lotka-Volterra systems possess
at least two distinct affine invariant lines ($x=0$ and $y=0$) and
the line at infinity is also invariant. We could also have other
invariant lines and each invariant line could have multiplicity
other than one.


\begin{definition} \label{def2.2} {\rm
Consider a planar quadratic differential system. We call
\emph{configuration of invariant lines} of this system and we
denote it by $\mathcal{C}$, the set of invariant lines (which may,
but not necessarily, have real coefficients) of the system, each
one which is  not filled up with singularities, endowed with its
own multiplicity and together with all the real  singular points
of this system, located on these invariant lines, each isolated
singularity endowed with its own multiplicity.  We denote by
$\mathcal{C}^*$ the set of all isolated invariant lines which are
not filled up with singularities. }
\end{definition}

This is a more powerful global classifying concept than anyone
used
 in \cite{Rey87}, \cite{Geo07}, \cite{CaoJia08}.

If a system has a finite number of invariant lines and each one of
them has finite multiplicity, we encode globally the information
regarding the multiplicities of the invariant lines of its
configuration in the notion of \emph{multiplicity divisor} of
invariant lines. Moreover we encode globally the information
regarding the multiplicities of the real singularities located on
the invariant lines in the configuration in the concept of
\emph{zero-cycle of multiplicities of singularities} of its
configuration. We have the following formal definitions:

\begin{definition} \label{def2.3} {\rm
 We consider an ${\rm LV}$-system possessing a
configuration $\mathcal{C}$ having a finite number of invariant
 lines not filled up with singularities, each with its
multiplicity.

(i) We attach to this system the \emph{multiplicity divisor on
the projective plane} corresponding to the configuration
 $\mathcal{C}$. This is defined as the formal sum:
\begin{equation}\nonumber
D_{\mathbb{C}}(\mathcal{C})=\sum_{L\in
\mathcal{C}^*}M(L)L,
\end{equation}
where $L$ is a projective invariant line of $\mathcal{C}$,
and $M(L)$ is the multiplicity of this line.

(ii) We attach to a configuration $\mathcal{C}$ the \emph{
multiplicity zero-cycle on the projective plane} which counts the
multiplicities of the real isolated singularities of the system
which are located on the configuration $\mathcal{C}$. This is the
formal sum:
\[
D_{\mathbb{R}}({\rm Sing},\mathcal{C})=\sum_{r\in
\mathcal{C}}m(r)r,
\]
where $m(r)$ is the multiplicity of the isolated singular point
$r$.

(iii) For a system \emph{(S)} with the line at infinity not
filled up with singularities we encode the multiplicities of
isolated singularities at infinity in the \emph{multiplicity
divisor on the line at infinity} which is the formal sum
\[
D_{\mathbb{C}}(\emph{S,Z})=\sum_{r\in \{Z=0\}}m(r)r,
\]
where $r$ is an isolated  singular point at infinity and $m(r)$
denotes its multiplicity. }
\end{definition}


We use the result which affirms that a quadratic Lotka-Volterra
differential system cannot have limit cycles. This theorem was
proved by Bautin in \cite{Bautin}.  Since this is an important
ingredient in determining all phase portraits of the
Lotka-Volterra systems we give here below its proof. Our proof is
a modification of  Coppel's proof in \cite{Cop} in order to make
the arguments more transparent by using a bit of Darboux theory
which enables us to effectively see the calculations.



\begin{theorem}[Bautin \cite{Bautin}]\label{thm:Bautin}
 The unique singular point inside
a periodic orbit of a Lotka-Volterra quadratic differential
systems is a center. Due to this   such a system is integrable via
the method of Darboux and so it has no limit cycle.
\end{theorem}

\begin{proof}
 Let $\gamma$ be a periodic orbit of a Lotka-Volterra
system. Since the two axes are affinely  invariant we may assume
that $\gamma$ is included in the interior of the first quadrant.
Let $p$ be the unique  singular point (see \cite{Cop}) inside
$\gamma$. The two axes $x=0$ and $y=0$ are invariant lines and
hence for any $\alpha,\beta$ in $\mathbb{C}$,
$R(x,y)=x^{\alpha}y^{\beta}=0$ is an invariant curve so we have
$\tilde DR=RK$ for
$$
K(x,y)=\alpha(a_0+a_1x+a_2y)+\beta(b_0+b_1x+b_2y)\in\mathbb{C}[x,y].
$$
To show that $p$ is a center it suffices to show that we can find
$\alpha,\beta \in \mathbb{C}$ such that $R$ is an integrating factor of
the system, i.e. $\frac{\partial(Rp)}{\partial
x}+\frac{\partial(Rq)}{\partial y}=0$. This means
$$
\frac{\partial(Rp)}{\partial x}+\frac{\partial(Rq)}{\partial y}
=\tilde DR+R\operatorname{div}(p,q)=R(K+\operatorname{div}(p,q))=0.
$$
Hence we search for $\alpha,\beta$ such that $K+\operatorname{div}(p,q)=0$.
This equation yields the system of equations:
\begin{equation}\label{sys:int-factor}
\begin{aligned}& \alpha a_0+\beta b_0=-a_0-b_0,\\
& \alpha a_1+\beta b_1=-2a_1-b_1,\\
&\alpha a_2+\beta b_2=-a_2-2b_2.\\
\end{aligned}\end{equation}
Since the singular point $p$ is isolated  and it is not on the
axes, $p$ is the unique solution of the equations:
$$
a_0+a_1x+a_2y=0, \quad  b_0+b_1x+b_2y=0
$$
and hence  $\widehat D=a_1b_2-a_2b_1\neq 0$. Therefore we can
solve the second and third equations in \eqref{sys:int-factor} in
$\alpha$ and $\beta$ and obtain
\begin{equation}\label{sys:2-3}
\alpha=-1+ b_2(b_1-a_1)/ \widehat D, \quad \beta=-1+
a_1(a_2-b_2)/\widehat D
\end{equation}
 Replacing this in $K+\operatorname{div}(p,q)$ we obtain
\begin{align*}
&K+ \operatorname{div}(p,q)\\
&=\alpha a_0+\beta b_0 +a_0+b_0\\
& = (-1+ b_2(b_1-a_1)/D)a_0+(-1+ (a_1(a_2-b_2))/\widehat D
b_0+a_0+b_0=g/\widehat D
\end{align*}
where
$$
g=a_0b_2(b_1-a_1)+a_1b_0(a_2-b_2).
$$  Hence $\operatorname{div}(Rp,Rq)=R(K+\operatorname{div}(p,q))=Rg/\widehat D$.
To show that
$\operatorname{div}(Rp,Rq)=0$ it suffices to show that $g=0$. Since $\gamma$ is a
periodic orbit we have:
$$
\int_{\gamma}(Rqdx-Rpdy)=\int_0^T(Rq\dot{x}-Rp\dot{y})dt=\int_0^T(Rqp-Rpq)dt=0
$$
where $T$ is the period of $\gamma$.  We now use the  formula of
Green
$$
{\int}_{   \overset{o}{\gamma}}\operatorname{div}(Rp,Rq)dxdy=\int_{\gamma}Rqdx-Rpdy=0,
$$
where ${\overset{o}{\gamma}}$ is the interior set of $\gamma$.
 But calculations give
$$
 {\int}_{   \overset{o}{\gamma}}\operatorname{div}(Rp,Rq)dxdy=(g/\widehat D){\int}_{ \overset{o}{\gamma}}Rdxdy=0.
$$
 Since $x>0$ and $y>0$ we must have
$R>0$ and hence $g=0$. But this gives $\operatorname{div}(Rp,Rq)=Rg/\widehat D=0$ so $R$ is an integrating factor and
therefore $p$ is a center. Furthermore since the system is
integrable on the complement in $\mathbb{R}^2$ of the union of the two
axes,  it has no limit cycle.
\end{proof}

The study of quadratic systems possessing invariant straight lines
began in \cite{SchVul04-QTDS} and was continued in
\cite{SchVul08-RMJM}, \cite{SchVul08-NATMA},  and
\cite{SchVul08-BASM}.  The four works jointly taken cover the full
study of quadratic differential systems possessing invariant lines
of at least four total multiplicity. Among these systems some but
not all, belong to the class LV and for these systems we
therefore already have their topological classification. We also
have the topological classification of all LV- systems with the
line at infinity filled up with singularities in
\cite{SchVul08-JDDE}.

To complete the    topological classification of all LV-systems
it thus suffices     to give a topological classification of:
\emph{a)} the class of LV-systems possessing exactly three
invariant lines all simple; \emph{b)}~the class of all
degenerate   LV-systems.



In \cite{SchVul10-JFPTA} all   possible 65 distinct configurations
of invariant lines of the LV-systems were listed and necessary
and sufficient conditions for the realization of each one of them
were given.  As we need these results we state them in the
Subsection \ref{subs: Prelim-results} below.
 The systems split into six
distinct classes according to the multiplicities of their
invariant lines (including the line at infinity).  The necessary
and sufficient conditions for the realization of each one of the
configurations are expressed in \cite{SchVul10-JFPTA} in terms of
invariant polynomials, with respect to the action of the affine
group and time homotheties.

\subsection{Group actions on polynomial systems}

Consider real planar polynomial  differential systems
\eqref{sys:PSgen}.   We   denote by PS the set of all planar
polynomial systems \eqref{sys:PSgen} of a fixed degree $n$.  On
the set PS acts (left action)  the group $\operatorname{Aff}(2,\mathbb{R})$ of
affine transformations on the plane:

\begin{equation}\label{map: group action}
 \begin{array}{rcl}
   \operatorname{Aff}(2,\mathbb{R})\times PS &\to & PS \\
   (g,\, S) &\to & \tilde S=gS
\end{array}
\end{equation}
This action is defined as follows:

Consider an affine transformation  $ g\in \operatorname{Aff}(2,\mathbb{R})$, $g:\
\mathbb{R}^{2}\to \mathbb{R}^{2}$.  For this
transformation we have:
$$
  g:\begin{pmatrix}\tilde x\\ \tilde y\end{pmatrix} =
      M\begin{pmatrix}x\\ y\end{pmatrix} +B;\quad
  g^{-1}:\begin{pmatrix} x\\ y\end{pmatrix}
 =      M^{-1}\begin{pmatrix}\tilde x\\ \tilde y\end{pmatrix} -M^{-1}B.
$$
where $M=\| M_{ij} \| $ is a $2\times 2$ nonsingular matrix and
$B$ is a $2\times 1$ matrix over $\mathbb{R}$. For every
$ S\in PS$ we can form its induced transformed system $\tilde S=g S$:
\begin{equation}
  \frac{d\tilde x}{dt} =\tilde p(\tilde x,\tilde y),\quad
  \frac{d\tilde y}{dt} =\tilde q(\tilde x,\tilde y),\label{Stilde}
\end{equation}
where
$$
 \begin{pmatrix} \tilde p(\tilde x,\tilde y)\\
\tilde q(\tilde x,\tilde y)\end{pmatrix} =
      M\begin{pmatrix} (p\circ {g^{-1}})(\tilde x,\tilde y)\\
                   (q\circ {g^{-1}})(\tilde x,\tilde y) \end{pmatrix}.
$$
The map \eqref{map: group action} verifies the axioms for a left
group action. For every  subgroup $G\subseteq \operatorname{Aff}(2,\mathbb{R})$
we have an  induced action of $G$ on PS.

\begin{definition} \rm
Consider a subset $\mathcal A$ of PS and a subgroup $  G$ of
$\operatorname{Aff}(2,\mathbb{R})$.  We say that the subset $\mathcal A$ is
invariant with respect to the group $  G$ if for every $g$ in $ G$
and for every system $S$ in $\mathcal A$  the transformed system
$gS$ is also in $\mathcal A$.
\end{definition}

We can identify the set  of systems in PS with a subset of 
$\mathbb{R}^m$  via the embedding  $PS \hookrightarrow\mathbb{R}^{m}$ 
which associates to  each system $(S)$ in PS the m-tuple 
$(a_{00},\ldots,b_{0n})$ of its coefficients.
 We denote by $\mathbb{R}^{m}_{\mathcal A}$
 the image of the subset $\mathcal A$ of PS under the embedding  $PS
 \hookrightarrow\mathbb{R}^{m}$.

 For every $ g\in \operatorname{Aff}(2,\mathbb{R})$ let $r_g:\
\mathbb{R}^{m}\to \mathbb{R}^{m}$ be the map which corresponds to $g$
via this action. We know \mbox{(cf. \cite {Sib88-Mon})} that $r_g$
is linear and that the map $r:\ \operatorname{Aff}(2,\mathbb{R})\to
GL(m,\mathbb{R})$ thus obtained is a group homomorphism. For every
subgroup $G$ of $\operatorname{Aff}(2,\mathbb{R})$, $r$ induces a representation of $G$
onto a subgroup $\mathcal G$ of $GL(m,\mathbb{R})$.

The group $\operatorname{Aff}(2,\mathbb{R})$ acts on QS and this yields an action of
this group on $\mathbb{R}^{12}$.  For every subgroup $G$ of
$\operatorname{Aff}(2,\mathbb{R})$, $r$ induces a representation of $G$ onto a subgroup
$\mathcal G$ of $GL(12,\mathbb{R})$.

\subsection{Definitions of invariant polynomials}

\begin{definition} \label{def:comit}\rm
 A polynomial $U( a\,,x,y)\in \mathbb{R}[a,x,y]$ is called a comitant
with respect to $(\mathcal{A},G)$,
where $\mathcal{A}$ is  an affine invariant subset of PS and $G$
is a subgroup  of $\operatorname{Aff}(2,\mathbb{R})$, if there exists $\chi\in
\mathbb{Z}$ such that for every \mbox{$(g, a )\in G\times\mathbb{R}^{m}_{\mathcal A} $} the following identity holds in $\mathbb{R}[x,y]$:
$$
U(r_g(a) , g(x,y)\,)\equiv\ (\det\,g)^{-\chi}\, U( a ,x,y),
$$
where $\det g=\det M$. If the polynomial $U$ does not explicitly
depend  on $x$ and $y$ then it is called invariant. The number $
\chi\in \mathbb{Z} $  is called the   {\sl weight\,} of the comitant $ U(
a ,x,y)$.
If $G=GL(2,\mathbb{R})$ (or $G=Aff(2,\mathbb{R})$\,)  and
$\mathcal{A}=PS$  then the comitant $U( a ,x,y)$   is called
$GL$-comitant (respectively,  affine comitant).
\end{definition}

\begin{definition}\label{def_G} \rm
A subset $X\subset \mathbb{R}^{m}$ will be called $G$-invariant,
if \ for every $ g\in G$ we have $r_g(X)\subseteq X$.
\end{definition}


Let $T(2,\mathbb{R})$ be the subgroup of $\operatorname{Aff}(2,\mathbb{R})$ formed by
translations. Consider the linear representation of $T(2,\mathbb{R})$ into
its corresponding subgroup ${\mathcal T}\subset GL(m,\mathbb{R})$,
i.e. for every $ \tau\in T(2,\mathbb{R})$, $\tau:\ x=\tilde x+\alpha,
y=\tilde y+\beta$ we consider as above $r_\tau:\
\mathbb{R}^{m}\to \mathbb{R}^{m}$.

\begin{definition} \rm
A  comitant $U( a ,x,y)$   with respect to $(\mathcal{A},G)$ is
called a $T$-comitant if for every $(\tau,  \mbox{\boldmath $a$}
)\in T(2,\mathbb{R})\times \mathbb{R}^{m}_{\mathcal A}$ the
identity\ $ U(r_\tau\cdot  \mbox{\boldmath $a$}, \tilde
x, \tilde y)\, =\, U( \mbox{\boldmath $a$} , \tilde x, \tilde
y) $ holds in $\mathbb{R}[ \tilde x,\tilde y]$.
\end{definition}


\begin{definition} \rm
 The polynomial $U( a ,x,y)\in \mathbb{R}[ a ,x,y]$
 has  well determined sign on \mbox{$V\subset\mathbb{R}^{m}$} with respect to $x, y$
 if for every fixed $a\in V$, the polynomial function
 $U(a,x,y)$ is not identically zero on $V$ and
 has constant sign outside its set of zeroes on $V$.
\end{definition}

\begin{observation}\rm
 We draw  attention to the fact, that if a T-comitant $U( a
,x,y)$ with respect to $(\mathcal A,G)$ of even weight is a binary
form in $x$, $y$, of even degree in the coefficients  of
\eqref{sys:PSgen} and has well determined sign on the affine
invariant algebraic subset $\mathbb{R}^{m}_{\mathcal A}$ then this
property is conserved by any affine transformation and the sign is
conserved.
\end{observation}


 \subsection{Main  invariant polynomials associated with
LV-systems}\label{subs: pol-inv}

 Consider real quadratic systems; i.e., systems  of
the form:
\begin{equation}\label{sys:QSgen}
 \begin{gathered}
\dot x =p_0+p_1(a,x,y)+\,p_2(a,x,y)\equiv p(a,x,y), \\
\dot y =q_0+ q_1(a,x,y)+\,q_2(a,x,y)\equiv q(a,x,y)
\end{gathered}
\end{equation}
with  $\max(\deg(p), \deg(q))=2$     and
\begin{gather*}
p_0=a_{00},\quad p_1(a,x,y)=  a_{10}x+ a_{01}y,\quad
p_2(a,x,y)= a_{20}x^2 +2
a_{11}xy + a_{02}y^2,\\
q_0=b_{00},\quad q_1(a,x,y)=  b_{10}x+ b_{01}y,\quad\ q_2(a,x,y)=
b_{20}x^2 +2
b_{11}xy + b_{02}y^2,
\end{gather*}
 where $
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})$
is the 12-tuple of the coefficients of an arbitrary system
\eqref{sys:QSgen}
 and denote
$$\mathbb{R}[
a,x,y]=\mathbb{R}[a_{00},a_{10},a_{01},a_{20},a_{11},a_{02},b_{00},b_{10},b_{01},
b_{20},b_{11},b_{02},x,y].
$$

\begin{notation} \rm
We denote by $a=(a_{00},a_{10}\ldots,b_{02})$ a
specific point in $\mathbb{R}^{12}$ and we keep $a_{ij}$ and $b_{ij}$ as
parameters. Each particular system \eqref{sys:QSgen} yields an
ordered 12-tuple $a$ of its coefficients.
\end{notation}


Let us consider the polynomials
\begin{equation}\label{expr:Ci-Di}
\begin{gathered}
 C_i(a,x,y)=yp_i(a,x,y)-xq_i(a,x,y)\in \mathbb{R}[a,x,y], i=0,1,2, \\
  D_i(a,x,y)=\frac{\partial}{\partial x}p_i(a,x,y)+
        \frac{\partial}{\partial y}q_i(a,x,y)\in \mathbb{R}[a,x,y], i=1,2.
\end{gathered}
\end{equation}
 As it was shown in \cite{Sib88-Mon} the polynomials
\begin{equation}\label{C_i:D_i}
\big\{\ C_0(a,x,y),\quad C_1(a,x,y),\quad C_2(a,x,y),\quad D_1(a),
\quad D_2(a,x,y)\ \big\}
\end{equation}
 of degree one in the coefficients of systems \eqref{sys:QSgen} are
$GL$-comitants   of these systems.

\begin{notation} \rm
Let $f, g\in \mathbb{R}[a,x,y]$ and
\begin{equation}\label{def:transv}
  (f,g)^{(k)}=
   \sum_{h=0}^k (-1)^h \begin{pmatrix}k\\ h\end{pmatrix}
   \frac{\partial^k f}{ \partial x^{k-h}\partial y^h}\
   \frac{\partial^k g}{ \partial x^h\partial y^{k-h}}.
\end{equation}
 $(f,g)^{(k)}\in \mathbb{R}[a,x,y] $ is called the transvectant of index
$k$ of $(f,g)$ (cf. {\rm\cite{GraYou41}, \cite{Olv99}})
\end{notation}

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

\begin{remark}   \rm
We point out that the elements of the set \eqref{C_i:D_i}
generate the whole set of $GL$-comitants and hence also the set of
affine comitants as well as of set of the $T$-comitants.
\end{remark}



\begin{notation}\label{not:1} \rm
 Consider the polynomial
$\Phi_{\alpha,\beta}=\alpha P+\beta Q\in \mathbb{R}[a,X,Y,Z,\alpha,\beta]$
where
$P=Z^2p(X/Z,Y/Z),$ $Q=Z^2q(X/Z,Y/Z)$, $p$,
 $q\in \mathbb{R}[a,x,y]$ and \\
$\max (\deg_{(x,y)}p,\deg_{(x,y)}q)=2$. Then
\begin{align*}
\Phi_{\alpha,\beta}&= c_{11}(a,\alpha,\beta)X^2  +2
c_{12}(a,\alpha,\beta)XY
       + c_{22}(a,\alpha,\beta)Y^2+
   2c_{13}(a,\alpha,\beta)XZ\\
 &\quad +2c_{23}(a,\alpha,\beta)YZ   +c_{33}(a,\alpha,\beta)Z^2
\end{align*}
and we denote
\begin{equation}\label{expr:D,H}
\begin{gathered}
  D(a,x,y) = 4\big[\det||c_{ij}(a,y,-x)   ||_{i,j\in\{1,2,3\}}\big],\\
 H(a,x,y)  =
4\big[\det||c_{ij}(a,y,-x) ||_{i,j\in\{1,2\}}\big].
\end{gathered}
\end{equation}
\end{notation}

We construct the following $T$-comitants:

\begin{notation}\label{not1} \rm
\begin{equation}\label{Comit:Bi}
\begin{gathered}
B_3(a,x,y)=(C_2,D)^{(1)}= \det\left[\operatorname{Jacobian}\left(C_2,D\right)\right],\\
B_2(a,x,y)=\left(B_3,B_3\right)^{(2)} - 6B_3(C_2,D)^{(3)},\\
B_1(a)=\operatorname{Res}_x\left( C_2,D\right)/y^9=-2^{-9}3^{-8}\left(B_2,B_3\right)^{(4)}.
\end{gathered}
\end{equation}
\end{notation}

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

 Let us consider the following $GL$-comitants of systems \eqref{sys:QSgen}:

\begin{notation}\label{not3} \rm
\begin{equation}\label{expr:M,K,N,eta,theta,mu0}
\begin{gathered}
   M(a,x,y) = (C_2,C_2)^{(2)}=2\,\operatorname{Hessian}\big(C_2(x,y)\big), \quad
    \eta(a) = \operatorname{Discrim}\big(C_2(x,y)\big),\\
   K(a,x,y) = \det\big[\operatorname{Jacobian}\big(p_2(x,y),q_2(x,y)\big)\big],\quad
   \mu_0(a) =  \operatorname{Discrim}\big(K(a,x,y)\big)/16,\\
 N(a,x,y) =  K(a,x,y) + H(a,x,y), \quad
   \theta(a)  =   \operatorname{Discrim}\big( N(a,x,y)\big).
\end{gathered}
\end{equation}
 \end{notation}

\begin{remark} \rm
 We note that by the discriminant   of the cubic form
$C_2(a,x,y)$ we mean the expression given in Maple via the
function ``${\rm discrim}(C_2,x)/y^6$''.
\end{remark}

  The geometrical meaning of these invariant polynomials is
revealed by the next 3 lemmas (see \cite{SchVul04-QTDS}).

\begin{lemma}\label{lem:Geom-mu0-K-H}
Let $(S)\in  QS$ and let $a\in \mathbb{R}^{12}$
be its 12-tuple of coefficients.  The common points of $P=0$ and
$Q=0$ ($P,Q$ are the homogenizations of $p,q$) on the line $Z=0$
are given by the common linear factors over $\mathbb{C}$ of $p_2$ and
$q_2$. This yields the geometrical meaning of the comitants
$\mu_0$, $K$ and $H$:
\begin{align*}
&\gcd (p_2(x,y),q_2(x,y)) \\
&=\begin{cases}
           \operatorname{constant} & \text{if }   \mu_0(a)\ne0;\\
           bx+cy & \text{if }   \mu_0(a)=0   \text{ and } K( a ,x,y)\ne0;\\
          (bx + cy)(dx + ey) & \text{if }   \mu_0(a) = 0, K( a ,x,y) = 0
                           \text{ and }   H( a ,x,y) \ne 0;\\
          (bx+cy)^2 & \text{if }   \mu_0(a) = 0, K( a ,x,y) = 0
                           \text{ and }   H( a ,x,y) = 0,
          \end{cases}
\end{align*}
where $bx+cy, dx+ey\in \mathbb C[x,y]$ are some linear forms and
$be-cd\ne0$.
\end{lemma}

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

\begin{lemma}\label{lem:S1-S5}
The form
of the divisor $D_{\mathbb{C}}(\emph{S,Z})$ for systems
\eqref{sys:QSgen} is determined by the corresponding conditions
indicated in Table 1, where we write
$\omega_1^c+\omega_2^c+\omega_3$ if two of the points, i.e.
$\omega_1^c, \omega_2^c$, are complex but not real.
\end{lemma}

\begin{table}[!htb]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\multicolumn{3}{c}{Table 1}\\[1mm]
\hline
  \raisebox{-0.7em}[0pt][0pt]{Case}  & \raisebox{-0.7em}[0pt][0pt]{Form of $D_{\mathbb{C}}(\emph{S,Z})$}
      & Necessary and sufficient   \\[-1mm]
         & & conditions on the comitants \\
 \hline\hline
 \rule{0pt}{1.2em} $1$ & $\omega_1+\omega_2+\omega_3 $ &  $\eta>0 $ \\[1mm]
\hline
 \rule{0pt}{1.2em}$2$  & $\omega_1^c+\omega_2^c+\omega_3 $ &  $\eta<0$ \\[1mm]
\hline
 \rule{0pt}{1.2em}  $3$ & $2\omega_1+\omega_2 $ &  $\eta=0,\quad M\ne0$ \\[1mm]
\hline
 \rule{0pt}{1.2em} $4$ & $3\omega $ &  $ M=0,\quad C_2\ne0$ \\[1mm]
\hline
 \rule{0pt}{1.2em} $5$ & $D_{\mathbb{C}}(\emph{S,Z})$ undefined  &  $ C_2=0$ \\[1mm]
\hline
\end{tabular}
\end{center}
\end{table}


To construct other necessary invariant polynomials let us
consider the differential operator ${\mathcal L}= x\cdot  L_2
-y\cdot L_1$ acting on $\mathbb{R}[a,x,y]$ constructed in
\cite{BalVul97-DEDS} (see also \cite{BalVul98-BASM}), where
\begin{gather*}
    L_1= 2a_{00}\frac{\partial}{\partial a_{10}} +
            a_{10}\frac{\partial}{\partial a_{20}} +
    \frac{1}{2}a_{01}\frac{\partial}{\partial a_{11}} +
            2b_{00}\frac{\partial}{\partial b_{10}} +
            b_{10}\frac{\partial}{\partial b_{20}} +
     \frac{1}{2}b_{01}\frac{\partial}{\partial b_{11}};\\
    L_2= 2a_{00}\frac{\partial}{\partial a_{01}} +
            a_{01}\frac{\partial}{\partial a_{02}} +
     \frac{1}{2}a_{10}\frac{\partial}{\partial a_{11}} +
            2b_{00}\frac{\partial}{\partial b_{01}} +
            b_{01}\frac{\partial}{\partial b_{02}} +
     \frac{1}{2}b_{10}\frac{\partial}{\partial b_{11}}.
\end{gather*}

In \cite{BalVul97-DEDS} it is shown that if a polynomial $U\in
\mathbb{R}[a,x,y]$ is a $GL$-comitant of system  \eqref{sys:QSgen}
then ${\mathcal L}(U)$ is also a $GL$-comitant.

By using this operator  and the $GL$-comitant
$\mu_0(a)=\operatorname{Res}_x(p_2(x,y),q_2(x,y))/y^4$ we   construct the
following polynomials:
\begin{equation}\label{expr:mu-i}
    \mu_i(a,x,y) =\frac{1}{i!} {\mathcal L}^{(i)}(\mu_0), \ i=1,..,4,
    \quad \text{where}\quad  {\mathcal L}^{(i)}(\mu_0)={\mathcal L}({\mathcal
    L}^{(i-1)}(\mu_0)).
\end{equation}
These polynomials are in fact $GL$-comitants of systems
\eqref{sys:QSgen}. The geometrical meaning of the $GL$-comitants
$\mu_i(a, x, y), i=0,1,\ldots,4 $   is revealed by the next 2
lemmas (see \cite{SchVul05-JDE}).

 \begin{lemma}\label{lm_1}
The system $P(X,Y,Z)=Q(X,Y,Z)=0$  possesses  exactly four solutions
counted with the multiplicities. Then $m$ $(1\le m\le4)$ of
these solutions lie on  $Z=0$   if and only if for every 
$i\in\{0,1,\ldots,m-1\}$ we have $\mu_i(a,x,y)=0$ 
and $\mu_m(a,x,y)\ne0$ as  polynomials in $\mathbb{R}[x,y]$.
\end{lemma}

\begin{lemma} \label{lem:Degener}
 A quadratic system
\eqref{sys:QSgen} is degenerate (i.e. $\gcd(p,q)\ne \operatorname{constant}$) if
and only if $\mu_i(a,x,y)=0$ as  polynomials in $\mathbb{R}[x,y]$
for every $i=0,1,2,3,4$.
\end{lemma}

Using the transvectant differential operator \eqref{def:transv}
and the invariant polynomials \eqref{expr:Ci-Di}, \eqref{expr:D,H}
and \eqref{expr:M,K,N,eta,theta,mu0} constructed earlier, we
define the following invariant polynomials which will be needed
later (see also \cite{SchVul08-NATMA}, \cite{SchVul08-BASM}):
\begin{gather*} H_1(a)= -\big((C_2,C_2)^{(2)},C_2)^{(1)},D\big)^{(3)};\\
  H_2(a,x,y)=\left(C_1, 2H-N\right)^{(1)}-2D_1N;\\
  H_3(a,x,y)=(C_2,D)^{(2)};\\
 H_4(a)=\big((C_2,D)^{(2)},(C_2,D_2)^{(1)}\big)^{(2)};\\
 H_5(a)=\big((C_2,C_2)^{(2)},(D,D)^{(2)}\big)^{(2)}+
    8\big((C_2,D)^{(2)},(D,D_2)^{(1)}\big)^{(2)};\\
  H_6(a,x,y)= 16N^2(C_2,D)^{(2)}+ H_2^2(C_2,C_2)^{(2)};\\
 H_7(a)=(N,C_1)^{(2)};\\
H_8(a)= 9\big((C_2,D)^{(2)},(D,D_2)^{(1)}\big)^{(2)}+
2\Big[(C_2,D)^{(3)}\Big]^2;\\
H_9(a)= -\Big(\big((D,D)^{(2)},D \big)^{(1)},D\Big)^{(3)};\\
H_{10}(a)= \big((N,D)^{(2)}, D_2\big)^{(1)};\\
H_{11}(a,x,y)=8 H\big[(C_2,D)^{(2)}+8(D,D_2)^{(1)}\big]+3H_2^2;\\
H_{12}(a,x,y)=(D,D)^{(2)}\equiv \operatorname{Hessian}(D);\\
H_{13}(a,x,y)=2 (\tilde
B,C_2)^{(3)}+\big((C_2,D)^{(2)}+(D_2,D)^{(1)},\tilde
E\big)^{(2)};\\
\begin{aligned}
H_{14}(a,x,y)
&= 96(D,C_2)^{(3)}(9\mu_0+\eta)\\
&\quad -4\Big(\Big(\big((B_3,D_2)^{(1)},D_2\big)^{(1)},
 D_2\Big)^{(1)},D_2\Big)^{(1)}
 -54\big((H,\tilde F)^{(1)},K\big)^{(2)}\\
&\quad -9\Big[\Big( \big(2(C_2,D)^{(2)}+11(D_2,D)^{(1)},H\big)^{(1)},K\Big)^{(2)}\Big];
\end{aligned}\\
N_1(a,x,y)=C_1(C_2,C_2)^{(2)} -2C_2(C_1,C_2)^{(2)};\\
N_2(a,x,y)=D_1(C_1,C_2)^{(2)}-\big((C_2,C_2)^{(2)},C_0\big)^{(1)};\\
N_5(a,x,y)= \big[(D_2,C_1)^{(1)} + D_1D_2\big]^2
-4\big(C_2,C_2\big)^{(2)}\big(C_0,D_2\big)^{(1)},\\
{\mathcal G}_2(a)=8H_8-9H_5,\\
{\mathcal G}_3(a)=(\mu_0-\eta)H_1-6\eta(H_4+12H_{10}),
\end{gather*}
where $\tilde B(a,x,y)$, $\tilde E(a,x,y)$ and $\tilde F(a,x,y)$
are defined on the page \pageref{express:tilde A,B,etc} below.

Apart from the  invariant polynomials constructed above, which in
fact are responsible for the configurations of the invariant lines
for the family LV-systems, we also need    polynomials for
distinguishing   phase portraits.

First  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}=\left(C_0,C_{1}\right)^{(1)},\quad
   T_{2}=\left(C_0,C_{2}\right)^{(1)},\quad
   T_{3}=\left(C_0,D_{2}\right)^{(1)},\\
   T_{4}=\left(C_{1},C_{1}\right)^{(2)},\quad
   T_{5}=\left(C_{1},C_{2}\right)^{(1)},\quad
   T_{6}=\left(C_{1},C_{2}\right)^{(2)},\\
   T_{7}=\left(C_{1},D_{2}\right)^{(1)},\quad
   T_{8}=\left(C_{2},C_{2}\right)^{(2)},\quad
   T_{9}=\left(C_{2},D_{2}\right)^{(1)}.
\end{gathered}
\end{equation}
Then we define  a family of $T$-comitants   expressed through
$C_i$ $(i=0,1,2)$ and $D_j$ $(j=1,2)$ (see \cite{BulCalTimVul96}):
\label{express:tilde A,B,etc}
\[
\tilde A= \left(C_{1},T_{8}-2T_{9}+D_{2}^2\right)^{(2)}/144,
\]
\begin{align*}
\widetilde B=&
\Big\{16D_{1}\left(D_{2},T_{8}\right)^{(1)}\left(3C_{1}D_{1}-2C_0D_{2}+4T_{2}\right)
 \\
   & +32C_0\left(D_{2}, T_{9}\right)^{(1)}\left(3D_{1}D_{2}-5T_{6}+9T_{7}\right)\\
   &+2\left(D_{2},  T_{9}\right)^{(1)}\left(27C_{1}T_{4}-18C_{1}D_{1}^2\right.
    \left. -32D_{1}T_{2}+32\left(C_0, T_{5}\right)^{(1)}\right.\big)\\
   &+6\left(D_{2},    T_{7}\right)^{(1)}\big[8C_0(T_{8}-12T_{9})
    -12C_{1}(D_{1}D_{2}+T_{7})+ D_{1}(26C_{2}D_{1}+32T_{5})\\
   &  + C_{2}(9T_{4}+96T_{3})\big]\\
   &+6\left(D_{2},
   T_{6}\right)^{(1)}\left[32C_0T_{9}-C_{1}(12T_{7}+52D_{1}D_{2})\right.
    \left.-32C_{2}D_{1}^2\right]\\
 &+48D_{2}\left(D_{2}, T_{1}\right)^{(1)}\left(2D_{2}^2-T_{8}\right)\\
   &-32D_{1}T_{8}\left(D_{2}, T_{2}\right)^{(1)}+9D_{2}^2T_{4}\left(T_{6}-2T_{7}\right)
    -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_{1}D_{2}-2C_{2}D_{1}
   \right)
    +6D_{1}D_{2}T_{4}\left(T_{8}-7D_{2}^2-42T_{9}\right)\\
   &+12D_{1}\left(C_{1},T_{8}\right)^{(1)}\left(T_{7}+2D_{1}D_{2}\right)
    +96D_{2}^2\left[D_{1}\left(C_{1}, T_{6}\right)^{(1)}+D_{2}\left(C_0, T_{6}\right)^{(1)}\right]\\
   &-16D_{1}D_{2}T_{3}\left(2D_{2}^2+3T_{8}\right)
    -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_{1}D_{2}T_{4}T_{9}\Big\} /(2^{8}3^{3}),\\
\end{align*}
\begin{gather*}
\begin{aligned}
\widetilde D=&\big[2C_0(T_{8}-8T_{9}-2D_{2}^2)+C_{1}(6T_{7}-T_{6}
   -\left(C_{1},T_{5}\right)^{(1)}\\
 &+6D_{1}(C_{1}D_{2}
   -T_{5})-9D_{1}^2C_{2}\big]/36,
\end{aligned}\\
\widetilde E=
\left[D_{1}(2T_{9}-T_{8})-3\left(C_{1},T_{9}\right)^{(1)}
      -D_{2}(3T_{7}+D_{1}D_{2})\right]/72,\\
\begin{aligned}
\widetilde F=&
\Big[6D_{1}^2(D_{2}^2-4T_{9})+4D_{1}D_{2}(T_{6}+6T_{7})
             + 48C_0\left(D_{2},T_{9}\right)^{(1)} -9D_{2}^2T_{4} + 288 D_{1}\widetilde E\\
     &-24\left(C_{2},\widetilde D\right)^{(2)} + 120\left(D_{2},\widetilde
             D\right)^{(1)}
               - 36C_{1}\left(D_{2},T_{7}\right)^{(1)}
   + 8D_{1}\left(D_{2},T_{5}\right)^{(1)}\Big]/144,
\end{aligned}\\
\widetilde K= (T_{8}+4T_{9}+4D_{2}^2)/72
          \equiv   \big(p_{2}(x,y),q_{2}(x,y)\big)^{(1)}/4,\\
\widetilde H= (-T_{8}+8T_{9}+2D_{2}^2)/72.
\end{gather*}
These polynomials in addition with \eqref{expr:Ci-Di} and
\eqref{expr:Ti} will serve as bricks in constructing algebraic
affine invariants for   systems \eqref{sys:QSgen}. Using these
bricks, the minimal polynomial basis of affine invariants up to
degree 12, containing 42 elements $A_1,\ldots,A_{42}$, was
constructed in \cite{BulCalTimVul96}.  The following are the
elements of this polynomial basis:
\begin{alignat*}{2}
& A_{1}= \tilde  A, && A_{22}=  \big[C_{2}, \widetilde D)^{(1)}, D_{2}\big)^{(1)},
  D_{2}\big)^{(1)},\\ &  && \quad  D_{2}\big)^{(1)}  D_{2}\big)^{(1)}/{1152},\\
& A_{2}= (C_{2}, \widetilde  D)^{(3)}/12,
&& A_{23}= \big[\widetilde F, \widetilde H)^{(1)}, \widetilde K\big)^{(2)}/8,\\
& A_{3}=  \big[ C_{2}, D_{2})^{(1)}, D_{2}\big)^{(1)},
D_{2}\big)^{(1)}/48,
&& A_{24}= \big[C_{2}, \widetilde D)^{(2)}, \widetilde K\big)^{(1)}, \widetilde H\big)^{(2)}/32,\\
& A_{4}= (\widetilde H, \widetilde H)^{(2)},
&& A_{25}= \big[\widetilde D, \widetilde D)^{(2)}, \widetilde E\big)^{(2)}/16,\\
& A_{5}=  (\widetilde H, \widetilde K)^{(2)}/2,
&& A_{26}= (\widetilde B, \widetilde D)^{(3)}/36,\\
& A_{6}=  (\widetilde E, \widetilde H)^{(2)}/2,
&& A_{27}= \big[\widetilde B, D_{2})^{(1)}, \widetilde H\big)^{(2)}/24,\\
& A_{7}= \big[ C_{2}, \widetilde E)^{(2)}, D_{2}\big)^{(1)}/8,
&& A_{28}= \big[C_{2},\widetilde K)^{(2)},\widetilde D\big)^{(1)},\widetilde E\big)^{(2)}/16,\\
& A_{8}= \big[\widetilde D, \widetilde H)^{(2)},
D_{2}\big)^{(1)}/8,
&& A_{29}= \big[\widetilde D, \widetilde F)^{(1)}, \widetilde D\big)^{(3)}/96,\\
& A_{9}  = \big[\widetilde D, D_{2})^{(1)}, D_{2}\big)^{(1)},
D_{2}\big)^{(1)}/48,
&& A_{30}= \big[C_{2},\widetilde D)^{(2)},\widetilde D\big)^{(1)},\widetilde D\big)^{(3)}/288,\\
& A_{10}= \big[\widetilde D, \widetilde K)^{(2)},
D_{2}\big)^{(1)}/8,
&& A_{31}= \big[\widetilde D,\widetilde D)^{(2)},\widetilde K\big)^{(1)},\widetilde H\big)^{(2)}/64,\\
& A_{11}= (\widetilde F, \widetilde K)^{(2)}/4,
&& A_{32}= \big[\widetilde D, \widetilde D)^{(2)}, D_{2}\big)^{(1)}, \widetilde H\big)^{(1)}, D_{2}\big)^{(1)}/64,\\
& A_{12}= (\widetilde F, \widetilde H)^{(2)}/4,
&& A_{33}= \big[\widetilde D, D_{2})^{(1)}, \widetilde F\big)^{(1)}, D_{2}\big)^{(1)}, D_{2}\big)^{(1)}/128,\\
& A_{13}= \big[C_{2}, \widetilde H)^{(1)}, \widetilde
H\big)^{(2)}, D_{2}\big)^{(1)}/24,
&& A_{34}= \big[\widetilde D, \widetilde D)^{(2)}, D_{2}\big)^{(1)}, \widetilde K\big)^{(1)}, D_{2}\big)^{(1)}/64,\\
& A_{14}= (\widetilde B, C_{2})^{(3)}/36,
&& A_{35}= \big[\widetilde D, \widetilde D)^{(2)}, \widetilde E\big)^{(1)}, D_{2}\big)^{(1)}, D_{2}\big)^{(1)}/128,\\
& A_{15}= (\widetilde E, \widetilde F)^{(2)}/4,
&& A_{36}= \big[\widetilde D,\widetilde E)^{(2)},\widetilde D\big)^{(1)},\widetilde H\big)^{(2)}/16,\\
& A_{16}= \big[\widetilde E, D_{2})^{(1)}, C_{2}\big)^{(1)},
\widetilde K\big)^{(2)}/16,
&& A_{37}= \big[\widetilde D,\widetilde D)^{(2)},\widetilde D\big)^{(1)},\widetilde D\big)^{(3)}/576,\\
& A_{17}= \big[\widetilde D,\widetilde
D)^{(2)},D_{2}\big)^{(1)},D_{2}\big)^{(1)}/64,
&& A_{38}= \big[C_{2},\widetilde D)^{(2)}, \widetilde D\big)^{(2)}, \widetilde D\big)^{(1)}, \widetilde H\big)^{(2)}/64,\\
& A_{18}= \big[\widetilde D,\widetilde F)^{(2)},
D_{2}\big)^{(1)}/16,
&& A_{39}= \big[\widetilde D,\widetilde D)^{(2)},\widetilde F\big)^{(1)},\widetilde H\big)^{(2)}/64,\\
& A_{19}= \big[\widetilde D,\widetilde D)^{(2)},\widetilde
H\big)^{(2)}/16,
&& A_{40}= \big[\widetilde D,\widetilde D)^{(2)},\widetilde F\big)^{(1)},\widetilde K\big)^{(2)}/64,\\
& A_{20}= \big[C_{2},\widetilde D)^{(2)}, \widetilde
F\big)^{(2)}/16,
&& A_{41}= \big[C_{2},\widetilde D)^{(2)}, \widetilde D\big)^{(2)}, \widetilde F\big)^{(1)}, D_{2}\big)^{(1)}/64,\\
& A_{21}= \big[\widetilde D, \widetilde D)^{(2)}, \widetilde
K\big)^{(2)}/16,
&& A_{42}= \big[\widetilde D,\widetilde F)^{(2)},\widetilde F\big)^{(1)},D_{2}\big)^{(1)}/16.
\end{alignat*}
In the above list, the bracket ``$[$'' is  a shorthand  to avoid
placing  up to five parenthesizes ``$($'' which otherwise would be
necessary.

Finally we construct the  affine  invariants which we need (see
also \cite{ArtLliVul08-IJBCh}):
\begin{gather*}
U_1(a)= A_1( A_{1}  A_{2}- A_{14}- A_{15}), \\
U_2(a)= -2A_2^2-2 A_{17}- 3A_{19}+6A_{21}, \\
U_3(a)= 6A_1^2-3 A_{8}+  A_{10}+ A_{11}-3 A_{12}, \\
U_4(a)=   A_{30}, \quad
G_9(a)=   (A_{4}+2A_{5})/4,
\end{gather*}
\begin{align*}
& W_3(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} +195 A_{19})\\
   &\quad +2 A_2^2( A_{10}+3 A_{11})+102 A_3(3A_{30} -14
          A_{29}) -63 A_6(17 A_{25}+ 30 A_{26})\\
   &\quad +3 A_{10}(14 A_{18} -118 A_{17}+ 153 A_{19} +120 A_{21}) +6 A_7( 329 A_{25}
         - 108 A_{26})\\
   &\quad +3 A_8(164 A_{18} + 153 A_{19} -442 A_{17})
           +9 A_{12}(2 A_{20} -160 A_{17} -2 A_{18} -59 A_{19}) \\
   &\quad +3 A_1(77 A_2A_{14}+235 A_2A_{15} - 54 A_{36})
           +18 A_{21}(21 A_9 -5 A_{11}) + 302 A_2A_{34}\\
   &\quad -366 A_{14}^2 -12 A_{15}(71 A_{14} + 80 A_{15})\big]/9,
\\
&W_4( a) \\
&= \big[1512A_{1}^2(A_{30} - 2A_{29})-648 A_{15}A_{26}
       +72 A_{1}A_{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*}


\subsection{Preliminary results involving the use of polynomial
invariants}\label{subs: Prelim-results}


We consider the family of real quadratic systems
\eqref{sys:QSgen}. We shall use the following lemma, which gives
the conditions on the coefficients of the systems
\eqref{sys:QSgen}  so that   the  origin of coordinates be a
center.  To do this we present the systems \eqref{sys:QSgen} with
$a_{00}=b_{00}=0$  in the following tensorial form (see
\cite{Sib88-Mon}):
\begin{equation}\label{sys:tens-form}
\begin{gathered}
   \dfrac{dx^j}{ dt} =  a^{j}_{\alpha}x^{\alpha} + a^{j}_{\alpha\beta}
    x^{\alpha}x^{\beta},\quad
(j,\alpha, \beta =1,2);\\
    a^{1}_1 =a_{10}, \quad a^{1}_2 =a_{01},\quad
    a^{1}_{11} =a_{20}, \quad   a^{1}_{22} =a_{02},  \\
    a^{2}_1 =b_{10}, \quad a^{2}_2 =b_{01}, \quad
    a^{2}_{11} =b_{20}, \quad   a^{2}_{22} =b_{02},  \\
    a^{1}_{12} =a^{1}_{21}=a_{11},\quad  a^{2}_{12} =a^{2}_{21}=b_{11}.
\end{gathered}
\end{equation}

\begin{lemma}[\cite{Sib88-Mon}] \label{lem:Sib}
The singular point $(0,0)$ of a quadratic system
\eqref{sys:tens-form}    is a center   if and only if $I_2<0$,
$I_1=I_6=0$ and one of the following sets of  conditions holds:
$$
(1)\ I_3=0;\quad( 2)\ I_{13}=0;\quad (3)\ 5I_3-2I_4=13I_3-10I_5=0,
$$
where
\begin{gather*}
I_{1} = a^\alpha_\alpha,\quad I_{2} = a^\alpha_\beta
a^\beta_\alpha,\quad I_{3} = a^\alpha_p a^\beta_{\alpha
q}a^\gamma_{\beta \gamma}\varepsilon^{pq},\quad
I_{4} = a^\alpha_p a^\beta_{\beta q}a^\gamma_{\alpha \gamma}\varepsilon^{pq},\\
I_{5} = a^\alpha_p a^\beta_{\gamma q}a^\gamma_{\alpha
\beta}\varepsilon^{pq},\quad I_{6} = a^\alpha_p a^\beta_\gamma
a^\gamma_{\alpha q}a^\delta_{\beta \delta}
        \varepsilon^{pq},\quad
I_{13} = a^\alpha_p a^\beta_{q r}a^\gamma_{\gamma
s}a^\delta_{\alpha \beta}
         a^\mu_{\delta \mu}\varepsilon^{pq}\varepsilon^{rs}.
\end{gather*}
and the unit bi-vector $\varepsilon^{pq}$ has the  coordinates:
$\varepsilon^{12}=-\varepsilon^{21}=1$,
$\varepsilon^{11}=\varepsilon^{22}=0$.
\end{lemma}

 Following \cite{SchVul10-JFPTA} we denote by ${\rm QSL}_{ i}$  the family of
all non-degenerate quadratic differential systems possessing
invariant straight lines (including the line at infinity not
filled up with singularities) of total multiplicity $i$ with
$i\in \{3,4,5,6\}$

   The following is a corollary of Lemma \ref{lem:Bi-0}.

\begin{corollary} \label{crl:2intersLines}
 A necessary condition for a  quadratic system
\eqref{sys:QSgen} to be in the class LV (i.e. to possess two
intersecting real  invariant affine lines) is that the condition
$B_2(a,x,y)=0$ be verified in $\mathbb{R}[x,y]$.
 \end{corollary}

According to \cite{SchVul04-QTDS} and \cite{SchVul08-NATMA} we
have:

\begin{lemma} \label{lem:6a}  If a quadratic
system $(S)$ corresponding to a point $a\in\mathbb{R}^{12}$
 belongs to the class ${\rm QSL}_{4}\cup {\rm QSL}_{5}\cup $ ${\rm QSL}_{6}$,
then for this system one of
the  following sets of conditions are satisfied in $\mathbb{R}[x,y]$,
respectively:
\\
$(S)\in {\rm QSL}_{4}$  $\Rightarrow$ either
$\theta(a)\ne0$ and $B_3(a,x,y)=0$, or $\theta(a)=0= B_2(a,x,y)$;
\\
$(S)\in {\rm QSL}_{5}$  $\Rightarrow$ either $\theta(a)=0=B_3(a,x,y)$,
 or $N(a,x,y)=0= B_2(a,x,y)$;
\\
$(S)\in {\rm QSL}_{6}$  $\Rightarrow$  $N(a,x,y)=0= B_3(a,x,y)$.
\end{lemma}

The  next theorem sums up several results in
\cite{SchVul10-JFPTA,SchVul08-RMJM,SchVul08-BASM,SchVul08-JDDE}.

\begin{theorem}\label{thm:classif-config}
There are  65 distinct configurations of
planar  quadratic differential LV-systems, given in
Fig. \ref{Fig:Config-InfNs} and
Fig. \ref{Fig:Config-FinNs}. The systems split into six
distinct classes according to the multiplicities of their
invariant lines (including the line at infinity) and to the
presence  of  lines filled up with singularities, as follows:

I. The  LV-systems with exactly three invariant straight lines
which are all simple. These have 13 configurations 
{Config. 3.j}, $j=1,2,\ldots,13$. The affine invariant necessary and
sufficient conditions for the realization of each one of these
configurations as well as its respective representative are
indicated in Table~2.

II. The  LV-systems with four   invariant straight lines counted
with multiplicity.  These  have 19 configurations
{Config. 4.j} with
$j\in\{1,3,4,5,9,10,11,12,16,\ldots,26\}$.
  The affine invariant necessary
and sufficient conditions for the realization of each one of these
configurations as well as the additional conditions for the
respective phase portraits given in
Fig. \ref{Fig:Ph-Port:4-6IL} are indicated in Table 3.

III. The  LV-systems with five   invariant straight lines counted
with multiplicity.  These have 11 configurations
{Config. 5.j} with $j\in\{1,3,7,8,11,12,13,14,17,18,19\}$.
 The affine invariant necessary and
sufficient conditions for the realization of each one of these
configurations as well as the additional conditions for the
respective phase portraits  given in
Fig. \ref{Fig:Ph-Port:4-6IL} are indicated in Table 3.

IV. The  LV-systems with six invariant straight lines counted
with multiplicity.  These have four configurations
{Config. 6.j} with $j\in\{1,5,7,8\}$.  The affine
invariant necessary and sufficient conditions for the realization
of each one of these configurations  as well as the additional
conditions for the respective phase portraits  given in
Fig. \ref{Fig:Ph-Port:4-6IL} are indicated in Table 3.


V. The  non-degenerate LV-systems with a line of singularities at
infinity. For these systems the condition $C_2=0$ holds and they
have four configurations {Config. $C_2$.j} with $j\in
\{1,3,5,7\}$.  The affine invariant necessary and sufficient
conditions for the realization of each one of these configurations
as well as the additional conditions for the respective five phase
portraits given in Fig. \ref{Fig:Pictures:C2-0} are
indicated in Table 3.

VI. The  degenerate LV-systems defined by the conditions
$\mu_i=0$, $i=0,1,\ldots,4$  possessing at least one affine line
filled with singularities. These have 14 configurations
{Config. LV$_d$.j} with $j\in\{1,\ldots,14\}$. The affine
invariant necessary and sufficient conditions for the realization
of each one of these configurations as well as its respective
representative are indicated in Table~4.
\end{theorem}


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

We first prove two Lemmas (\ref{lem:M4-center} and \ref{lem:Types of FSP}) 
which will be  needed later.

We shall denote by $\tilde s$ (respectively $\tilde n$;
$\tilde f$; $\tilde c$; $\tilde s\tilde n$) a singular point
of \emph{saddle} (respectively \emph{node};
 \emph{focus}; \emph{center}; \emph{saddle-node}) type.

Assume that a quadratic system is an LV-system.
Then   due to an affine transformation we can assume that
this system belongs to the family of the systems
\begin{equation}\label{SLV:Generic}
  \dot x= x(c+gx+hy),\quad  \dot y=y(f+mx+ny),
\end{equation} which in the generic case  possess   the following finite
singularities: the origin which we denote by $M_1$, a singularity
on the $x$-axis which we denote by  $M_3$, another one on  the
$y$-axis denoted by $M_3$ and a fourth singularity denoted by
$M_4$ which is not located on anyone of the two axes. The specific
values od the coordinates are:
\begin{equation}\label{sing:Mi-generic}
 M_1(0,0),  \quad  M_2(-c/g,0),  \quad M_3(0,-f/n),  \quad
M_4\left(\frac{cn-fh}{hm-gn},  \frac{fg-cm}{hm-gn}\right)
\end{equation}
in case when the corresponding denominators are different from
zero.

The systems \eqref{SLV:Generic} possess in the generic case three
infinite singular points which are:
$$
 R_1(g-m, n-h, 0),\quad R_2(1,0, 0),\quad   R_3(0, 1, 0).
$$


\begin{observation}\label{obs:keep-index} \rm
The lines $y=0$ and $x=0$ intersect the
line at infinity at $(1, 0, 0)$ and $(0, 1, 0)$) respectively. We
denote by $R_2$ and $R_3$ these points so as to have $M_2$ and
$R_2$ (respectively $M_3$ and $R_3$) on the same invariant line.
\end{observation}

\begin{notation}\label{not:keep-Mi,Rj} \rm
Whenever for a system \eqref{SLV:Generic} we have a multiple
singularity we indicate which of  the singularities coalesced,for
example $M_3\equiv M_{1}$ or $M_4\equiv R_{1}$.
\end{notation}

For each finite singular point $M_j$, $j\in\{1,2,3,4\}$ we
denote its basic invariants: the trace $\rho_j$, the determinant
$\Delta_j$ and the discriminant $\delta_j$.

Similarly for each infinite singular point $R_j$, $j\in\{1,2,3\}$
we denote the respective basic invariants by $\tilde\rho_j$,
$\tilde\Delta_j$ and $\tilde\delta_j$.

 \begin{lemma}\label{lem:M4-center}
 Assume that  a quadratic system belongs to the
family  {\rm LV}.  Then $B_3=0$ is a necessary condition for
the system to have a center. Moreover, if this system possesses a
focus, then this focus could only be a strong one, i.e. its trace
is not zero.
\end{lemma}

\begin{proof} Since the points $M_1, M_2$ and $M_3$ are placed on the
invariant lines it is clear that  only the point $M_4$ (which
exists if $hm-gh\ne0$) could be of the focus-center type. As it is
known this occurs only if the discriminant $\delta_4$  of the
equation for the eigenvalues  corresponding to the point $M_4$ is
negative. Moreover it could be a center only if the corresponding
trace $\rho_4=0$, where
$$
\begin{gathered}\rho_4= \big[n(c\,m-fg) + g(fh-cn))\big]/(gn-hm),\\
  \delta_4= \left\{\big[n(c\,m-fg) - g(fh-cn))\big]^2+4hm(c\,m-fg)(fh-cn) \right\}/(gn-hm)^2 .
\end{gathered}$$
On the other hand for systems \eqref{SLV:Generic} calculation
yields
$$
B_3= 3(f-c)\big[n(c\,m-fg) +
g(fh-cn))\big]x^2y^2=3(f-c)\rho_4x^2y^2
$$
and therefore the condition $\rho_4=0$ implies $B_3=0$, i.e. the
last condition is necessary for the existence of a center.

Suppose now that a system from the family {\rm LV}-systems
possesses a weak focus. For this it is necessary to have
$\delta_4<0$ and $\rho_4=0$. But as we show below a singular point
satisfying these conditions is necessarily a center. Indeed
placing the singular point $M_4$ at the origin of coordinates we
get the systems
\begin{equation}\label{SLV:M4}
  \dot x= \Big(\frac{cn-fh}{hm-gn}+x\Big)(gx+hy),\quad
  \dot y=\Big(\frac{fg-cm}{hm-gn}+y\Big)(mx+ny).
\end{equation}   Considering Lemma \ref{lem:Sib} for these systems
calculations yield
\begin{equation}\label{val:I1-I6}
\begin{gathered}
I_1=\frac{ n(fg-c\,m) + g(cn-fh)}{hm-gn},\quad
2I_2=I_1^2+\delta_4,\\
I_3=  I_1 (mn-gh) /2,\quad
 I_6= \frac{I_1}{4(hm-gn)}W(c,f,g,h,m,n),
\end{gathered}
\end{equation}
 where $W(c,f,g,h,m,n)$ is a polynomial in the coefficients of
systems.  According to Lemma~\ref{lem:Sib} the condition $I_1=0$
is necessary for $(0,0)$ to be a center. Then $I_6=I_3=0$ and the
condition $\delta_4<0$ implies $I_2<0$. Thus the conditions of
Lemma \ref{lem:Sib} are satisfied and hence, $(0,0)$ of systems
\eqref{SLV:M4} and, consequently, the singular point $M_4$ of
systems \eqref{SLV:Generic} is a center.

Thus a Lotka-Volterra quadratic system could not possess a weak
focus.
\end{proof}

\begin{table}[!htb]
\footnotesize
\begin{tabular}{| l@{}|@{}c@{}|@{}c@{}|}
\multicolumn{3}{c}{Table 2}\\[1mm]
\hline \hfil \begin{tabular}{c}Orbit representative\end{tabular}   &
\begin{tabular}{c} Necessary and sufficient\\ conditions:\ $B_2=0$ and\end{tabular} & \begin{tabular}{c} Configu-\\ ration \end{tabular}\\
  \hline\hline\rule[0mm]{0mm}{10.0mm}
 ({\it III.1})\ $\left\{\!\!\begin{array}{l} \dot x= x [1   +  gx  + (h-1)y ], \\[-0.1mm]
                \dot y= y [f  +  (g-1)x  + hy ],\\[-0.2mm]
                f,g,h\in\mathbb{R},\quad \text{cond.}\  (\mathcal{A}_1)     \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta>0,\
                 \mu_0B_3H_9\ne0\ \text{and either}\\ \theta\ne0\ \text{or}\ (\theta=0\ \&\ NH_7\ne0) \end{array} $  & \text{Config. 3.1}\\  \hline
 ({\it III.2})\ $\left\{\!\!\begin{array}{l} \dot x= x [1   + gx  + (h-1)y ], \\[-0.1mm]
                \dot y= y [  (g-1)x  + hy ],\\[-0.2mm]
                g,h\in\mathbb{R},\quad \text{cond.}\  (\mathcal{A}_2) \end{array}\!\!\right.$ &  $ \!\!\!
                \begin{array}{c}\eta>0,\,
                 \mu_0 B_3\ne0,\, H_9=H_{13}=0\\  \text{and}\  \text{either}\ \theta\ne0\\ \text{or}\ (\theta=0\ \&\ NH_7\ne0) \end{array} $  & \text{Config. 3.2}\\ \hline
  ({\it III.3})\ $\left\{\!\!\begin{array}{l} \dot x= x [g   + gx  + (h-1)y ], \\[-0.1mm]
                \dot y= y [g-1  +  (g-1)x  + hy ],\\[-0.2mm]
                g,h\in\mathbb{R},\quad \text{cond.} \  (\mathcal{A}_2)  \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta>0,\
                 \mu_0 B_3H_{13}\ne0,\ H_9=0 \\  \text{and}\  \text{either}\ \theta\ne0\\ \text{or}\ (\theta=0\ \&\ NH_7\ne0)\end{array} $  & \text{Config. 3.3}\\ \hline
  ({\it III.4})\ $\left\{\!\!\begin{array}{l} \dot x= x [1  + (h-1)y ], \\[-0.1mm]
                \dot y= y(f - x+hy),\\[-0.2mm]
                f,h\in\mathbb{R},\quad \text{cond.}\  (\mathcal{A}_3)  \end{array}\!\!\right.$ &  $
                \begin{array}{c}\eta>0,
                \theta B_3H_9\ne0,  \mu_0= H_{14}=0 \end{array} $  & \text{Config. 3.4}\\ \hline
  ({\it III.5})\ $\left\{\!\!\begin{array}{l} \dot x= x [1  + (1-h)(x-y)], \\[-0.1mm]
                \dot y= y(f- hx+hy),\\[-0.2mm]
                 f,h\in\mathbb{R},\quad \text{cond.}\  (\mathcal{A}_3)  \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta>0,\
                \theta B_3H_9H_{14} \ne0,\ \mu_0 =0 \end{array} $  & \text{Config. 3.5}\\ \hline
  ({\it III.6})\ $\left\{\!\!\begin{array}{l} \dot x= x [1  + (h-1)y ], \\[-0.1mm]
                \dot y= y(-x+hy),\\[-0.2mm]
                h\in\mathbb{R},\, h(h - 1) \ne\!0  \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta>0,\
                \theta B_3\ne0,\ \mu_0= H_9=0,\\  H_{13}=H_{14}=0 \end{array} $  & \text{Config. 3.6}\\ \hline
  ({\it III.7})\ $\left\{\!\!\begin{array}{l} \dot x=  x [h-1  + (h-1)y ], \\[-0.1mm]
                \dot y= y(h-x+hy),\\[-0.2mm]
                h\in\mathbb{R},\, h(h - 1) \ne 0  \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta>0,\
                \theta B_3H_{13}\ne0,\\ \mu_0=  H_9 = H_{14} =0 \end{array} $  & \text{Config. 3.7}\\ \hline
  ({\it III.8})\ $\left\{\!\!\begin{array}{l} \dot x= x [1  + (1-h)(x-y)], \\[-0.1mm]
                \dot y= hy(y- x),\\[-0.2mm]
                 h\in\mathbb{R},\, h(h - 1) \ne 0  \end{array}\!\!\right.$ &  $
                 \begin{array}{c}\eta>0,
                \theta B_3 H_{14} \ne0,  \mu_0 =H_9=0 \end{array} $  & \text{Config. 3.8}\\ \hline
  ({\it III.9})\ $\left\{\!\!\begin{array}{l} \dot x= x(1+ gx+ y), \\[-0.1mm]
                \dot y= y(f-x+gx+y),\\[-0.2mm]
                 f,g\in\mathbb{R},\quad \text{cond.}\  (\mathcal{A}_4)  \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta=0,\
                \theta H_4 B_3 \mu_0 H_9 \ne0  \end{array} $  & \text{Config. 3.9}\\ \hline
  ({\it III.10})\ $\left\{\!\!\begin{array}{l} \dot x= x(g+ gx+ y), \\[-0.1mm]
                \dot y= y[g-1+(g-1)x+y],\\[-0.2mm]
                  g\in\mathbb{R},\quad g(g-1)\ne0  \end{array}\!\!\right.$ &  $
                  \begin{array}{c}\eta=0,
                \theta H_4 B_3 \mu_0H_{13} \ne0, H_9=  0  \end{array} $  & \text{Config. 3.10}\\ \hline
  ({\it III.11})\ $\left\{\!\!\begin{array}{l} \dot x= x(1+ gx+ y), \\[-0.18mm]
                \dot y= y(-x+gx+y),\\[-0.2mm]
                  g\in\mathbb{R},\quad g(g-1)\ne0  \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta=0,\
                \theta H_4 B_3 \mu_0  \ne0,\\ H_9=H_{13}=0  \end{array} $  & \text{Config. 3.11}\\ \hline
  ({\it III.12})\ $\left\{\!\!\begin{array}{l} \dot x= x(1+  y), \\[-0.8mm]
                \dot y= y(f+x+y),\\[-0.2mm]
                  f\in\mathbb{R},\, f(f\!-\!1)\!\ne\!0  \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta=0,\
                \theta H_4 B_3 H_9 \ne0,\ \mu_0=0  \end{array} $  & \text{Config. 3.12}\\ \hline
  ({\it III.13})\ $\left\{\!\!\begin{array}{l} \dot x= x(1+  y), \\[-0.8mm]
                \dot y= y(x+y),\\[-0.8mm]
                  \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta=0,\,
                \theta H_4 B_3  \ne0,\, \mu_0=H_9=0  \end{array}\!\! $  &\, \text{Config. 3.13}\\   \hline
  \multicolumn{2}{|c }{\rule[0mm]{0mm}{5.0mm}$gh(g+h-1)(g-1)(h-1)f(f-1)(fg+h)(1-g+fg)(f+h-fh)\ne0;$} & $(\mathcal{A}_1) $\\[1.8mm]
  \multicolumn{2}{|c }{$gh(g+h-1)(g-1)(h-1)\ne0;$} & $(\mathcal{A}_2) $\\[1.8mm]
  \multicolumn{2}{|c }{$h(h-1)f(f-1)(f+h-fh)\ne0.$} &  $(\mathcal{A}_3) $\\[1.8mm]
  \multicolumn{2}{|c }{$g(g-1)f(f-1)(1-g+fg)\ne0.$} &  $(\mathcal{A}_4) $\\[1.8mm]
 \hline
\end{tabular}
\end{table}

\begin{table}[!htb]
\begin{center}
{\scriptsize
\begin{tabular}{|@{}c@{}|@{}c@{}|@{}c@{}| c |}
\multicolumn{4}{c}{ Table 3 }\\[1mm]
\hline $\begin{array}{c}\text{Configuration}\end{array}$  &
$\begin{array}{c}\text{Necessary and sufficient}\\
\textit{conditions}
\end{array}$ & $\begin{array}{c}\textit{Additional conditions}\\ \textit{for phase
portraits}
\end{array}$ & $\begin{array}{c}\text{Phase}\\ \text{portrait}\end{array}$ \rule{0pt}{6.7mm}\\[0.4mm]
\hline
  \raisebox{-1.5em}[0pt][0pt]{\text{Config. 4.1}} &  \raisebox{-1.5em}[0pt][0pt]{$\begin{array}{c}\eta>0,\
               B_3=0,\  \theta\ne0,\  H_7\ne0\end{array} $} & $\mu_0>0$ & \text{Picture 4.1(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $\mu_0<0,\ K<0$ &\text{Picture 4.1(b)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $\mu_0<0,\ K>0$ &\text{Picture 4.1(c)}\rule{0pt}{3.8mm}\\[0.3mm]
 \hline
  \raisebox{-1.5em}[0pt][0pt]{\text{Config. 4.3}} &  \raisebox{-1.5em}[0pt][0pt]{$\begin{array}{c}\eta>0,\
                B_3=0,\  \theta\ne0,\\  H_7=0,\ H_1\ne0,\ \mu_0\ne0 \end{array} $} & $\mu_0>0$ & \text{Picture                 4.3(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $\mu_0<0,\ K<0$ &\text{Picture 4.3(b)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $\mu_0<0,\ K>0$ &\text{Picture 4.3(c)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
  \raisebox{-0.7em}[0pt][0pt]{\text{Config. 4.4}} &  \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}\eta>0,\
                 B_3=0,\  \theta\ne0,\\ H_7=0,\ H_1\ne0,\ \mu_0=0 \end{array} $} & $K<0$ & \text{Picture                 4.4(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $ K>0$ &\text{Picture 4.4(b)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
  \raisebox{-1.5em}[0pt][0pt]{\text{Config. 4.5}} &  \raisebox{-1.5em}[0pt][0pt]{$\begin{array}{c}\eta>0,\
                 B_3=0,\  \theta\ne0,\\ H_7=0,\  H_1 =0 \end{array} $} & $\mu_0>0$ & \text{Picture                 4.5(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $\mu_0<0,\ K<0$ &\text{Picture 4.5(b)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $\mu_0<0,\ K>0$ &\text{Picture 4.5(c)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
  \raisebox{-2.2em}[0pt][0pt]{\text{Config. 4.9}} &
  \raisebox{-2.2em}[0pt][0pt]{$\begin{array}{c}\eta>0,\
                 B_2= \theta=H_7=0,\\[0.4mm] \mu_0B_3H_4H_9\ne0\ \ \textit{and\ either}\\[0.4mm]
                  H_{10}N>0\ \ \textit{or}\ \ N=0,\ H_8>0  \end{array} $}  & ${\mathcal G}_2>0,H_4>0,{\mathcal G}_3<0$ & \text{Picture                 4.9(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & ${\mathcal G}_2 < 0$  &\raisebox{-0.7em}[0pt][0pt]{\text{Picture 4.9(b)}}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-3} & & ${\mathcal G}_2>0,H_4<0$ & \rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & ${\mathcal G}_2>0,H_4>0,{\mathcal G}_3>0$ &\text{Picture 4.9(c)}\rule{0pt}{3.8mm}\\[0.3mm]
 \hline
  \raisebox{-3.1em}[0pt][0pt]{\text{Config. 4.10}} &  \raisebox{-1.5em}[0pt][0pt]{$\begin{array}{c}\eta>0,\
                 B_3\ne0,\ B_2= \theta=0,\\ \mu_0\ne0,  H_7=H_9=0,  H_{10}N>0   \end{array} $}
                  & $H_4>0,{\mathcal G}_3>0$ & \text{Picture                 4.10(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $H_4<0$ &\text{Picture 4.10(b)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $H_4>0,{\mathcal G}_3<0$ &\raisebox{-1.2em}[0pt][0pt]{\text{Picture 4.10(c)}}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{2-3} &\ $\begin{array}{c}\eta>0,B_3H_4\ne0,\\ B_2=N=H_9=0,H_8>0\end{array}$\,& -- & \rule{0pt}{3.8mm}\\[0.3mm]
\hline
  \raisebox{-0.7em}[0pt][0pt]{\text{Config. 4.11}} &
  \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}\eta=0,\ MB_3\ne0,\
               B_2=  \theta=0,\\ H_7=0,\ \mu_0\ne0,\ H_{10}>0 \end{array} $} & $H_4>0$ & \text{Picture                 4.11(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $ H_4<0$ &\text{Picture 4.11(b)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
  \raisebox{-2.7em}[0pt][0pt]{\text{Config. 4.12}} &
  \raisebox{-2.7em}[0pt][0pt]{$\begin{array}{c}\eta=0,\ M\ne0,\
               B_3=  \theta=0,\\ KH_6\ne0,  H_7=\mu_0=0,  H_{11}>0 \end{array} $} & $\mu_2>0,\ L>0$ & \text{Picture                 4.12(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $\mu_2>0,\ L<0$ &\text{Picture 4.12(b)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $\mu_2<0,\ K<0$ &\text{Picture 4.12(c)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $\mu_2<0,  K>0,  L>0$ &\text{Picture 4.12(d)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $\mu_2<0, K>0,  L<0$ &\,\text{Picture 4.12(e)}\,\rule{0pt}{3.8mm}\\[0.3mm]
\hline
  \raisebox{-0.7em}[0pt][0pt]{\text{Config. 4.16}} &  \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}\eta>0,\
               B_3\ne0,\ B_2= \theta=0,\\ \mu_0=H_7=0,\ H_9\ne0 \end{array} $} & ${\mathcal G}_2>0$ & \textit{Portrait
                4.16(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & ${\mathcal G}_2<0$ &\textit{Portrait 4.16(b)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 4.17}} &
  \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\eta>0,\
              B_3\ne0,\ B_2=\theta=0,\\ \mu_0=H_7=H_9=0,\ H_{10}\ne0 \end{array} $} & -- & \text{Picture                 4.17}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
  \raisebox{-0.7em}[0pt][0pt]{\text{Config. 4.18}} &  \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}\eta>0,\
               B_3= \theta=0,\\ \mu_0=0,\ H_7\ne0 \end{array} $} & $\mu_2L>0$ & \text{Picture                 4.18(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $\mu_2L<0$ &\text{Picture 4.18(b)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
  \raisebox{-0.7em}[0pt][0pt]{\text{Config. 4.19}} &  \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}\eta=0,
              M\ne0, B_3=\theta=K=0,\\ NH_6\ne0,\ \mu_0=H_7=0,\ H_{11}\ne0 \end{array} $} & $\mu_3K_1<0$ & \text{Picture                 4.19(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $\mu_3K_1>0$ &\text{Picture 4.19(b)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
  \raisebox{-0.7em}[0pt][0pt]{\text{Config. 4.20}} &
  \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}\eta=0,\
              M\ne0,\ B_3=0,\ \theta\ne0,\\ H_7=0,\ D=0 \end{array} $} & $\mu_0>0$ & \text{Picture                 4.20(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $\mu_0<0$ &\text{Picture 4.20(b)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
\end{tabular}
}\end{center}

\end{table}
\begin{table}[!htb]
\begin{center}
{\scriptsize
\begin{tabular}{|@{}c@{}|@{}c@{}| c |@{}c@{}|}
\multicolumn{4}{c}{ Table 3 (continued)}\\[1mm]
\hline $\begin{array}{c}\text{Configuration}\end{array}$  &
$\begin{array}{c}\text{Necessary and sufficient}\\
\textit{conditions}
\end{array}$ & $\begin{array}{c}\text{Additional}\\ \text{conditions for }\\ \textit{phase
portraits}
\end{array}$ & $\begin{array}{c}\text{Phase}\\ \text{portrait}\end{array}$ \rule{0pt}{6.7mm}\\[0.4mm]
\hline\rule{0pt}{5.2mm}
  \raisebox{-0.7em}[0pt][0pt]{\text{Config. 4.21}} &
  \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}\eta=0,\
              M\ne0,\ B_3=0,\ \theta\ne0,\\ H_7=0,\ D\ne0,\ \mu_0\ne0 \end{array} $} & $\mu_0>0$ & \text{Picture                 4.21(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $\mu_0<0$ &\text{Picture 4.21(b)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
  \raisebox{-1.5em}[0pt][0pt]{\text{Config. 4.22}} &  \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}\eta>0,\
                 B_3\ne0,\ B_2= \theta=0,\\ \mu_0\ne0,\ N\ne0,\ H_7=H_{10}=0 \end{array} $}
                  & $H_1>0$ &\, \text{Picture                 4.22(a)} \rule{0pt}{3.8mm}\\[0.3mm]
  \cline{3-4} & & $H_1<0$ &\raisebox{-0.7em}[0pt][0pt]{\text{Picture 4.22(b)}}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{2-3} &\ $\eta>0,B_3H_4\ne0, B_2=\theta=N=H_8=0$\,& -- & \rule{0pt}{3.8mm}\\[0.3mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 4.23}} &
  \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\eta=0,\
              MB_3\ne0,\ B_2=\theta=0,\\ \mu_0\ne0,\ H_7= H_{10}=0 \end{array} $} & -- & \text{Picture 4.23}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
  \raisebox{-0.7em}[0pt][0pt]{\text{Config. 4.24}} &
  \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}\eta=0,
              M\ne0, B_3=\theta=0,\\ KH_6\ne0,\ \mu_0=H_7= H_{11}=0 \end{array} $} & $L>0$ & \text{Picture                 4.24(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $L<0$ &\text{Picture 4.24(b)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
  \raisebox{-0.7em}[0pt][0pt]{\text{Config. 4.25}} &
  \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}\eta=0,\
              M\ne0,\ B_3=0,\ \theta\ne0,\\ H_7\ne0 \end{array} $} & $\mu_0>0$ & \text{Picture                 4.25(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $\mu_0<0$ &\text{Picture 4.25(b)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 4.26}} &
  \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\eta=0,\
              M\ne0,\ B_3=0,\ \theta\ne0,\\ H_7=0,\ D\ne0,\ \mu_0=0 \end{array} $} & -- & \text{Picture                 4.26}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 5.1}} &
  \raisebox{-0.0em}[0pt][0pt]{$ \begin{array}{c}\eta>0,\ B_3=\theta=0,\\
   N\ne0,\, \mu_0\ne0,\, H_1\ne0\end{array} $} & -- & \text{Picture    5.1}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 5.3}} &
  \raisebox{-0.0em}[0pt][0pt]{$ \begin{array}{c}\eta>0,\, B_2=N=0,\, B_3\ne0,\\
   H_1>0, \  H_4=0,\  H_5>0 \end{array} $} & -- & \text{Picture    5.3}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 5.7}} &
  \raisebox{-0.0em}[0pt][0pt]{$ \begin{array}{c}\eta>0,\, B_3=\theta=0,\\[-0.8mm]
            N\ne0,\, \mu_0= H_6=0\end{array} $} & -- & \text{Picture    5.7}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 5.8}} &
  \raisebox{-0.0em}[0pt][0pt]{$ \begin{array}{c} \eta>0,\, B_3=\theta=0,\\
            N\ne0,\, \mu_0\ne0,\, H_1=0 \end{array} $} & -- & \text{Picture    5.8}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 5.11}} &
  \raisebox{-0.0em}[0pt][0pt]{$ \begin{array}{c} \eta=0,\,M\ne0,\, B_3=\theta=0,\\
             \mu_0\ne0,\, N\ne0,\,D\ne0 \end{array} $} & -- & \text{Picture    5.11}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 5.12}} &
  \raisebox{-0.0em}[0pt][0pt]{$ \begin{array}{l} \eta>0,\, B_2=N=0,\,
             B_3\ne0,\\ H_1>0,\,  H_4= H_5=0 \end{array} $} & -- & \text{Picture    5.12}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 5.13}} &
  \raisebox{-0.0em}[0pt][0pt]{$ \begin{array}{c} \eta=0,\,M\ne0, B_3=N=0,\\
              H=N_1=0,\, N_2D\ne0,\, N_5>0 \end{array} $} & -- & \text{Picture    5.13}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
  \raisebox{-0.7em}[0pt][0pt]{\text{Config. 5.14}} &  \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c} \eta=0,\,M\ne0, B_3=\theta=0,\\
             NK\ne0,\ \mu_0=H_6=0 \end{array}$} & $L>0$ & \text{Picture 5.14(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-4} & & $L<0$ &\text{Picture 5.14(b)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 5.17}} &
  \raisebox{-0.0em}[0pt][0pt]{$ \begin{array}{c} \eta=0,\,M\ne0, B_3=N=0,\\
  H=N_1=N_5=0,\,   N_2D\ne0 \end{array} $} & -- & \text{Picture    5.17}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 5.18}} &
  \raisebox{-0.0em}[0pt][0pt]{$ \begin{array}{c} \eta=0,\,M\ne0, B_3=\theta=0,\\
  N\ne0,\, \mu_0=K=H_6=0   \end{array} $} & -- & \text{Picture    5.18}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 5.19}} &
  \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c} \eta=0,\,M\ne0, B_3=\theta=0,\\
  \mu_0\ne0,\, N\ne0,\,D=0  \end{array}$} & -- & \text{Picture    5.19}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 6.1}} &
  \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{l}\eta>0,\ B_3= N=0,\ H_1>0 \end{array} $} & -- &
  \text{Picture    6.1}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 6.5}} &
  \raisebox{-0.0em}[0pt][0pt]{$ \begin{array}{l}\eta>0,\ B_3= N=\ H_1=0 \end{array} $} & -- &
  \text{Picture    6.5}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
%  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 6.7}} &
%  \raisebox{-0.0em}[0pt][0pt]{ $ \begin{array}{c} MD\ne0,\eta=\ B_3= N=0,\\ H=N_1=N_2=0\end{array} $ } & -- &
%  \text{Picture    6.7}\rule{0pt}{5.9mm}\\[1.8mm]
%
%\hline
\end{tabular}
}\end{center}
\end{table}



\begin{table}[!htb]
\begin{center}
{\scriptsize
\begin{tabular}{|@{}c@{}|@{}c@{}| c | c  |}
\multicolumn{4}{c}{ Table 3(continued)}\\[1mm]
\hline $\begin{array}{c}\text{Configuration}\end{array}$  &
$\begin{array}{c}\text{Necessary and sufficient}\\
\textit{conditions}
\end{array}$ & $\begin{array}{c}\textit{Additional}\\ \textit{conditions for }\\ \textit{phase
portraits}
\end{array}$ & $\begin{array}{c}\text{Phase}\\ \text{portrait}\end{array}$ 
\rule{0pt}{6.7mm}\\[0.4mm]
\hline\rule{0pt}{5.2mm}
%
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 6.7}} &
  \raisebox{-0.0em}[0pt][0pt]{ $ \begin{array}{c} MD\ne0,\eta=\ B_3= N=0,\\ H=N_1=N_2=0\end{array} $ } & -- &
  \text{Picture    6.7}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
%
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. 6.8}} &
  \raisebox{-0.0em}[0pt][0pt]{ $\begin{array}{c} MH\ne0,\ \ \eta= B_3= N=0,\\ H_2=0, H_3>0 \end{array}$ } & -- &
  \text{Picture    6.8}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. $C_2.$1}} &
  \raisebox{-0.0em}[0pt][0pt]{ $\begin{array}{c} C_2=0,\, H_{10}\ne0,\ H_{9}<0\end{array}  $ } & -- &
  \text{Picture    $C_2.$1}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. $C_2.$3}} &
  \raisebox{-0.0em}[0pt][0pt]{ $\begin{array}{c} C_2=0,\, H_{10}\ne0, H_{9}=0, H_{12}\ne0\end{array}  $ } & -- &
  \text{Picture    $C_2.$3}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
  \raisebox{-0.7em}[0pt][0pt]{\text{Config. $C_2.$5}} &
  \raisebox{-0.7em}[0pt][0pt]{ $\begin{array}{c} C_2=0,\,  H_{10}=0,  H_{12}\ne0,  H_{11}>0\end{array}  $ } & $\mu_2<0$ &
  \text{Picture    $C_2.$5(a)}\rule{0pt}{5.9mm}\\[1.8mm]
 \cline{3-4} & & $\mu_2>0$ &\text{Picture  $C_2.$5(b)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
  \raisebox{-0.0em}[0pt][0pt]{\text{Config. $C_2.$7}} &
  \raisebox{-0.0em}[0pt][0pt]{ $\begin{array}{c} C_2=0,\,   H_{10}=0,  H_{12}\ne0,  H_{11}=0\end{array}  $ } & -- &
  \text{Picture    $C_2.$7}\rule{0pt}{5.9mm}\\[1.8mm]
\hline
\end{tabular}
}\end{center}
\end{table}


\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.95\textwidth]{fig1} % Portraits-LV-C2-0
\end{center}
\caption{Phase portraits  of LV-systems with all
points at infinity singular}\label{Fig:Pictures:C2-0}
\end{figure}


 \begin{figure}[htb]
\begin{center}
\includegraphics[width=0.95\textwidth]{fig2}{fig2} % Config-LV-InfNS
\end{center}
\caption{Configurations of LV-systems with infinite
number of singularities} \label{Fig:Config-InfNs}
\end{figure}


\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.95\textwidth]{fig3} % Portraits-LV-4-6ILs-c
\end{center}
\caption{Phase portraits of LV-systems with at least
4 invariant lines}\label{Fig:Ph-Port:4-6IL}
\end{figure}


\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.95 \textwidth]{fig4} % Config-LV-FinNS
\end{center}
\caption{Configurations of LV-systems with finite number
of singularities}\label{Fig:Config-FinNs}
\end{figure}


\begin{table}[!htb]
\begin{center}
{\footnotesize
\begin{tabular}{|l@{}|@{}c@{}|c|}
\multicolumn{3}{c}{Table 4} \\[3mm]
\hline  \begin{tabular}{c}Orbit representative\end{tabular}   &
\begin{tabular}{c} Necessary and sufficient\\ conditions\end{tabular} & \begin{tabular}{c} Configuration \end{tabular}\\
  \hline\hline\rule[0mm]{0mm}{9.0mm}
  ({\it LV$_d$.1})\ $\left\{\!\!\begin{array}{l} \dot x= x(1+gy-y), \\[-0.1mm]
                \dot y= (g-1)xy,\\[0.0mm]
                  g(g-1)\ne0\end{array}\!\!\right.$ &  $  \begin{array}{c}\eta>0,\
                  \mu_{0,1,2,3,4}=0,\, \theta \ne0,\ H_7\ne0  \end{array} $  &
                  \text{Config. LV$_d$.1}\\ [4.8mm]
\hline\rule[0mm]{0mm}{8.5mm}
  ({\it LV$_d$.2})\ $\left\{\!\!\begin{array}{l} \dot x= x(gx-y), \\[-0.1mm]
                \dot y= (g-1)xy,\\[-0.0mm]
                  g(g-1)\ne0 \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta>0,\
                  \mu_{0,1,2,3,4}=0,\, \theta \ne0,\ H_7=0  \end{array} $  & \text{Config. LV$_d$.2}\\ [4.8mm]
\hline\rule[0mm]{0mm}{6.0mm}
  ({\it LV$_d$.3})\ $\left\{\!\!\begin{array}{l} \dot x= x(1+y), \\[-0.1mm]
                \dot y= xy,\\[-0.0mm]
                  \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta>0,\
                  \mu_{0,1,2,3,4}=0,\, \theta =0,\\ H_4=0,\,H_7\ne0  \end{array} $  & \text{Config. LV$_d$.3}\\ [3.0mm]
\hline\rule[0mm]{0mm}{6.0mm}
  ({\it LV$_d$.4})\ $\left\{\!\!\begin{array}{l} \dot x= xy, \\[-0.1mm]
                \dot y= xy,\\[-0.0mm]
                  \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta>0,\
                  \mu_{0,1,2,3,4}=0,\, \theta =0,\\ H_4=0,\,H_7=0  \end{array} $  & \text{Config. LV$_d$.4}\\ [3.0mm]
 \hline\rule[0mm]{0mm}{6.0mm}
  ({\it LV$_d$.5})\ $\left\{\!\!\begin{array}{l} \dot x= xy, \\[-0.1mm]
                \dot y= y(1-x+y),\\[0.0mm]
                 \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta=0,\
                  \mu_{0,1,2,3,4}=0,\, \theta \ne0,\ H_7\ne0  \end{array} $  & \text{Config. LV$_d$.5}\\ [3.0mm]
\hline\rule[0mm]{0mm}{6.0mm}
  ({\it LV$_d$.6})\ $\left\{\!\!\begin{array}{l} \dot x= xy, \\[-0.1mm]
                \dot y= y(-x+y),\\[-0.0mm]
                 \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta=0,\
                  \mu_{0,1,2,3,4}=0,\, \theta \ne0,\ H_7=0  \end{array} $  & \text{Config. LV$_d$.6}\\ [3.0mm]
 \hline\rule[0mm]{0mm}{6.0mm}
  ({\it LV$_d$.7})\ $\left\{\!\!\begin{array}{l} \dot x= x(1+gx),  \\[-0.1mm]
                \dot y= (g-1)xy, \\[0.0mm]
                  g(g-1)\ne0\end{array}\!\!\right.$ &  $  \begin{array}{c}\eta=0,\
                  \mu_{0,1,2,3,4}=0,\, \theta=0,\\ K\ne0,\  H_2\ne0  \end{array} $  & \text{Config. LV$_d$.7}\\ [3.0mm]
\hline\rule[0mm]{0mm}{6.0mm}
  ({\it LV$_d$.8})\ $\left\{\!\!\begin{array}{l} \dot x= gx^2,   \\[-0.1mm]
                \dot y= (g-1)xy,\\[-0.0mm]
                  g(g-1)\ne0\end{array}\!\!\right.$ &  $  \begin{array}{c}\eta=0,\
                  \mu_{0,1,2,3,4}=0,\, \theta=0,\\ K\ne0,\  H_2=0 \end{array} $  & \text{Config. LV$_d$.8}\\ [3.0mm]
\hline\rule[0mm]{0mm}{6.0mm}
  ({\it LV$_d$.9})\ $\left\{\!\!\begin{array}{l} \dot x= x, \\[-0.1mm]
                \dot y=  xy,\\[-0.0mm]
                 \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta=0,\
                  \mu_{0,1,2,3,4}=0,\, \theta=0,\\ K=0,\,  N\ne0,\,H_7=0,\, H_2\ne0 \end{array} $  & \text{Config. LV$_d$.9}\\ [3.0mm]
\hline\rule[0mm]{0mm}{6.0mm}
  ({\it LV$_d$.10})\ $\left\{\!\!\begin{array}{l} \dot x=0, \\[-0.1mm]
                \dot y=  xy,\\[-0.0mm]
                 \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta=0,\
                  \mu_{0,1,2,3,4}=0,\, \theta=0,\, K=0,\\  N\ne0,\,H_7=0,\, H_2=0,\, D=0 \end{array} $  & \text{Config. LV$_d$.10}\\ [3.0mm]
\hline\rule[0mm]{0mm}{6.0mm}
  ({\it LV$_d$.11})\ $\left\{\!\!\begin{array}{l} \dot x=x(x+2), \\[-0.1mm]
                \dot y=  0,\\[-0.0mm]
                 \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta=0,\
                  \mu_{0,1,2,3,4}=0,\, \theta=0,\\ K=0,\,N=0,\,D=N_1=0,\, N_5>0 \end{array} $  & \text{Config. LV$_d$.11}\\ [3.0mm]
\hline\rule[0mm]{0mm}{6.0mm}
  ({\it LV$_d$.12})\ $\left\{\!\!\begin{array}{l} \dot x=x^2, \\[-0.1mm]
                \dot y=  0,\\[-0.0mm]
                 \end{array}\!\!\right.$ &  $  \begin{array}{c}\eta=0,\
                  \mu_{0,1,2,3,4}=0,\, \theta=0,\\ K=0,\,N=0,\,D=N_1=0,\, N_5=0 \end{array} $  & \text{Config. LV$_d$.12}\\ [3.0mm]
\hline\rule[0mm]{0mm}{6.0mm}
  ({\it LV$_d$.13})\ $\left\{\!\!\begin{array}{l} \dot x= x(1+x), \\[-0.1mm]
                \dot y=  xy,\\[-0.0mm]
                 \end{array}\!\!\right.$ &  $  \begin{array}{c}C_2=0,\
                  \mu_{0,1,2,3,4}=0,\,   H_2\ne0 \end{array} $  & \text{Config. LV$_d$.13}\\ [3.0mm]
\hline\rule[0mm]{0mm}{6.0mm}
  ({\it LV$_d$.14})\ $\left\{\!\!\begin{array}{l} \dot x= x^2, \\[-0.1mm]
                \dot y=  xy,\\[-0.0mm]
                 \end{array}\!\!\right.$ &  $  \begin{array}{c}C_2=0,\
                  \mu_{0,1,2,3,4}=0,\,   H_2=0 \end{array} $  & \text{Config. LV$_d$.14}\\ [3.0mm]
\hline
\end{tabular}
}\end{center}
\end{table}

   As it will follow from the proof of the Main Theorem
the following assertion is valid:

\begin{lemma}\label{lem:Types of FSP}
Assume that a quadratic system belongs
to the class LV. Then the system   possesses   one of the
following 18 configurations of finite singularities and each one
of them is realizable in this class:

\begin{tabular}{lllll}
  $(a)$\ \emph{$\tilde s,\tilde s,\tilde s,\tilde n$};
 & $(\stackrel{*}{a})$\ \emph{$\tilde s,\tilde s,\tilde s,\tilde f$};
 & $(b)$\ \emph{$\tilde s,\tilde n,\tilde n,\tilde n$};
 & $(\stackrel{*}{b})$\ \emph{$\tilde s,\tilde n,\tilde n,\tilde f$};
 & $(c)$\ \emph{$\tilde s,\tilde s,\tilde n,\tilde n$}; \\
  $(\stackrel{*}{c})$\ \emph{$\tilde s,\tilde s,\tilde n,\tilde f$};
 & $(d)$\ \emph{$\tilde s\tilde n,\tilde s,\tilde s$};
 & $(e)$\ \emph{$\tilde s\tilde n,\tilde n,\tilde n$};
 &  $(\stackrel{*}{e})$\ \emph{$\tilde s\tilde n,\tilde n,\tilde f$};
 & $(f)$\ \emph{$\tilde s\tilde n,\tilde s,\tilde n$}; \\
  $(\stackrel{*}{f})$\ \emph{$\tilde s\tilde n,\tilde s,\tilde f$};
 & $(g)$\ \emph{$\tilde s,\tilde s,\tilde n$};
 & $(\stackrel{*}{g})$\ \emph{$\tilde s,\tilde s,\tilde f$};
 & $(h)$\ \emph{$\tilde s,\tilde n,\tilde n$};
 & $(\stackrel{*}{h})$\ \emph{$\tilde s,\tilde n,\tilde f$}; \\
  $(k)$\ \emph{$\tilde s\tilde n,\tilde s$};
 & $(l)$\ \emph{$\tilde s\tilde n,\tilde n$};
 & $(\stackrel{*}{l})$\ \emph{$\tilde s\tilde n,\tilde f$}.
\end{tabular}
\end{lemma}

\begin{notation}\label{not:(a),(b)...}\rm
 For the notations of the phase portraits corresponding to
{Configs. 3.j} ($j=1,2,\ldots,13$) we shall use the number
\emph{3.j} of the configuration and the corresponding additional
couple $(ik)$ (or  $(\stackrel{*}{i}k)$) with
$$
(i)\in\{(a),(b),(c),(d),(e),(f),(g),(h),(k),(l)\}.
$$
The symbol $(i)$ depends on the configuration of the finite singularities
indicated by Lemma \ref{lem:Types of FSP},  whereas
$k\in\{1,\ldots,5\}$  indicates the number we give to the
respective phase portraits with the {Configs. 3.j}. For
example, the notation {Picture 3.3(e2)} denotes one of the
phase portraits associated with {Configs. 3.3} having the
finite singularities $\tilde s\tilde n,\tilde n,\tilde n$. We note
that we keep the same letter, only adding a star (i.e.
$(\stackrel{*}{i})$) in the case when one node is substituted by a
focus (which are locally topologically equivalent).
\end{notation}

\subsection*{Proof of the main theorem}
 To prove the main theorem we first notice that we can split the class of
all LV-systems into six distinct subclasses: \text{(i)} the class of
all LV-systems having exactly three simple real invariant lines;
\text{(ii)} the three classes of LV- systems possessing
invariant lines of total multiplicity 4, respectively 5 and 6;
\text{(iii)} the class of all   LV  systems with the line at
infinity filled up with singularities; \text{(iv)} the class of
all  LV  systems which are degenerate. We recall that in
\cite{SchVul08-RMJM} and \cite{SchVul08-BASM} the phase portraits
of the quadratic systems with invariant lines of total
multiplicity at least four are constructed. Moreover in
\cite{SchVul08-JDDE}  the topological classification  of   the
whole family of quadratic systems with the infinite line filled up
with singularities (the case $C_2=0$) is done and hence the phase
portraits for the cases \text{(ii)} and \text{(iii)} are
already done. So it remains firstly  to examine the cases
\text{(i)} which have the configurations given by
{Configs. 3.j} with $j=1,2,\ldots,13$  (see Subsection
\ref{subs:phase portraita-3 ILs}) and the case \text{(iv)} of
the degenerate LV-systems with the configurations given by
{Configs. LV$_d$.j} with $j=1,2,\ldots,14$ (see Subsection
\ref{subs:Phase-Portraits-Degen-LV}); and secondly to prove in
Subsection \ref{sec:topol-distinct} the statements \text{(v)} of
the main theorem.

\subsection{Phase portraits of LV-systems with exactly three simple
real invariant straight lines}\label{subs:phase portraita-3 ILs}

The   result concerning the  normal forms, the configurations and
the respective invariant criteria for this class is encapsulated
in Table 2. In this Table we observe that for any system with the
configuration of invariant lines given by {Configs.\,3.j}
($j=1,2,\ldots,13$) the condition $B_3\ne0$ holds. Therefore by
Lemma \ref{lem:M4-center} this system could not have a center.


\begin{theorem}
The phase portraits of Lotka-Volterra quadratic differential
systems possessing exactly three real invariant straight lines all
simple, correspond to 13 configurations. Adding up the numbers of
topologically distinct phase portraits for each of the 13
configurations we end up with a total of 65 phase portraits
(Fig. \ref{Fig:Ph-Port-3ILs}), only 60 of which are
topologically distinct. In Table 2 are listed in columns 2 and 3
the necessary and sufficient conditions for the realization of
each one of the portraits appearing in column four.
\end{theorem}


\begin{proof} We shall consider step by step each one of the
configurations {Configs. 3.j} ($j=1,2,\ldots,13$). For all
the configurations the proof follows the same pattern, which we
describe in the steps below.
\begin{itemize}
\item We take for a configuration {Configs.\,3.j}  its normal
form from Table 2 and we calculate the coordinates of its
singularities finite and infinite.
\item We evaluate for each finite singularity $M_i$ the   basic
invariants: the trace $\rho_i$, the determinant $\Delta_i$ and the
discriminant $\delta_i$.
\item For each infinite singularity $R_k$ it suffices to evaluate
 the determinant $\tilde \Delta_k$ and the trace $\tilde\rho_k$.
\item In order to use Table 1 from \cite{ArtLliVul08-IJBCh} we
evaluate for the normal form the invariant polynomials which we
need and which occur in the third column of this table. We
translate the inequalities given by these polynomials  into
inequalities of the coefficients.
\item We relate the signs of the invariant polynomials to the
signs of the basic invariants $\Delta_j$, $\tilde \Delta_k$ and
$\delta_j$. These give us the number of saddles and anti-saddles.
\item In order to determine the types of the anti-saddles, we
evaluate for the normal form the invariant polynomials we need and
which occur  in column four of Table 1 in
\cite{ArtLliVul08-IJBCh}. The inequalities involved in that
column, translate  into inequalities of the coefficients of the
normal form.
\item We use the invariant conditions in column 5 of Table 5 from
this article and we show that they lead to the phase portrait
given in column 6 of Table 5.
\end{itemize}
\end{proof}

\subsubsection{The phase portraits associated with {Config. 3.1}}

According to Table 2
we  consider the family of systems
\begin{equation}\label{sys:Config.3.1-G}
\dot x = x [1   +  gx  + (h-1)y ], \quad \dot y  = y [f  + (g-1)x + hy ],
\end{equation}
for which the condition
\begin{equation}\label{cond:A1}
gh(g+h-1)(g-1)(h-1)f(f-1)(fg+h)(g-1-fg)(fh-f-h)\ne0
\end{equation} holds. For  all four distinct finite singularities of systems
\eqref{sys:Config.3.1-G} with condition \eqref{cond:A1}, we
have
\begin{equation}\label{expr:Delta-i,delta-i:3.1}
\begin{gathered}
M_1(0,0):\quad \Delta_1=f, \quad \rho_1=f+1,\quad  \delta_1=(f-1)^2;\\
M_2(-1/g,0):\quad \Delta_2=(g-1-fg)/g, \quad \delta_2=(1+fg)^2/g^2;\\
M_3(0,-f/h):\quad \Delta_3=f(fh-f-h)/h, \quad \delta_3=(f+h)^2/h^2;\\
M_4\left(\frac{fh-f-h}{g+h-1},  \frac{g-1-fg}{g+h-1}\right):
\quad \Delta_4=\frac{(fh-f-h)(g-1-fg)}{g+h-1},\\
\quad   \rho_4=\frac{(fg+h)}{1-g-h},\quad \delta_4=\rho_4^2-4\Delta_4.
\end{gathered}
\end{equation}
 and for the three infinite singular points we obtain
\begin{equation}\label{ISPs:tilde delta-i:3.1}
 R_1(1, 1, 0):\  \widetilde{\Delta}_1= 1 -g- h ;\quad
  R_2(1, 0, 0): \  \widetilde{\Delta}_2 = g;\quad
   R_3(0, 1, 0): \ \widetilde{\Delta}_3 = h.
\end{equation}


\begin{table}[!htb]
\begin{center}
{\footnotesize
\begin{tabular}{|c|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|}
\multicolumn{6}{c}{Table 5}\\[1mm]
\hline $ \begin{gathered}{Configu-}\\
\emph{ration}\end{gathered} $  & $\begin{gathered}\emph{Necessary and suffi-}\\
\emph{cient conditions}
\end{gathered}$ & \multicolumn{3}{c|}{$ \begin{gathered}\emph{Additional conditions}\\ \emph{for phase
portraits}\end{gathered}$} & $\begin{gathered}\emph{Phase}\\ \emph{portrait}\end{gathered} $ \rule{0pt}{6.7mm}\\[0.4mm]
\hline
 \multirow{40}*{{Config. 3.1}} &   \multirow{40}*{$\begin{gathered}\eta>0,
                 \mu_0B_3H_9\ne0,\\ B_2=0\ \emph{and}\\ \emph{either}\ \  \theta\ne0\ \emph{or}\\ (\theta=0\ \&\ NH_7\ne0) \end{gathered}$}
    & \multirow{5}*{$\begin{gathered}\mu_0<0,\\  K<0\end{gathered}$}& \multirow{3}*{$W_4\ge0$} &$ B_3U_1 < 0, U_{2} < 0$ &{Picture
                3.1(a1)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & &\,$ B_3U_1 < 0, U_{2} > 0\,$  &{Picture
                3.1(a2)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & &\,$ B_3U_1  > 0\,$ &{Picture
                3.1(a3)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{4-6}
         & & &  \multirow{0}*{\,$W_4<0$\,} &$ -$ &{Picture 3.1($ \stackrel{\,*}{a} $2)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{3-6}
         & & \multirow{20}*{$\begin{gathered}\mu_0<0,\\  K>0\end{gathered}$}& \multirow{10}*{$\begin{gathered}W_4>0\ \emph{or}\\  W_4=0\ \&\\ W_3\ge0 \end{gathered}$ }
         &$ B_3U_1 < 0$ &\raisebox{-1.1em}[0pt][0pt]{{Picture  3.1(b1)}}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-5}
         & & & & $\begin{gathered} B_3U_1> 0, U_{2} > 0,\\ U_{4} > 0,U_{3}> 0\end{gathered}$  & \rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & & $\begin{gathered} B_3U_1> 0, U_{2} < 0,\\ B_{3}H_{14}> 0\end{gathered}$  &\raisebox{-1.1em}[0pt][0pt]{{Picture  3.1(b2)}} \rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-5}
         & & & & $\begin{gathered} B_3U_1> 0, U_{2} > 0,\\ U_{4} > 0,U_{3}< 0\end{gathered}$  & \rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & & $\begin{gathered} B_3U_1> 0, U_{2} < 0,\\ B_{3}H_{14}< 0\end{gathered}$  &\raisebox{-1.1em}[0pt][0pt]{\,{Picture  3.1(b3)}\,} \rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-5}
         & & & & $\begin{gathered} B_3U_1> 0, U_{2} > 0,\\ U_{4} < 0\end{gathered}$  & \rule{0pt}{3.8mm}\\[0.3mm]
\cline{4-6}
         & &  & \multirow{9}*{$\begin{gathered}W_4<0\ \emph{or}\\  W_4=0\ \&\\ W_3<0 \end{gathered}$ }
         &$ B_3U_1 < 0,U_2<0$ &\raisebox{-1.1em}[0pt][0pt]{{Picture  3.1($\stackrel{*}{b}$1)}}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-5}
         & & & & $\begin{gathered} B_3U_1> 0, U_{2} > 0,\\ U_{4} > 0,U_{3}> 0\end{gathered}$  & \rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & & $\begin{gathered} B_3U_1> 0, U_{2} < 0\end{gathered}$  &\raisebox{-1.1em}[0pt][0pt]{{Picture  3.1($\stackrel{*}{b}$2)}} \rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-5}
         & & & & $\begin{gathered} B_3U_1> 0, U_{2} > 0,\\ U_{4} > 0,U_{3}< 0\end{gathered}$  & \rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & &  $\begin{gathered} B_3U_1> 0, U_{2} > 0,\\ U_{4} < 0\end{gathered}$   &\raisebox{0em}[0pt][0pt]{\,{Picture  3.1($\stackrel{*}{b}$3)}\,} \rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & &  $\begin{gathered} B_3U_1< 0, U_{2} > 0 \end{gathered}$   &\raisebox{0em}[0pt][0pt]{\,{Picture  3.1($\stackrel{*}{b}$4)}\,} \rule{0pt}{3.8mm}\\[0.3mm]
\cline{3-6}
         & & \multirow{15}*{$\begin{gathered}\mu_0>0\end{gathered}$}& \multirow{10}*{$\begin{gathered}W_4>0\ \emph{or}\\  W_4=0\ \&\\ W_3\ge0 \end{gathered}$ }
         &$ U_2<0,B_3H_{14} < 0$ &\raisebox{-1.1em}[0pt][0pt]{{Picture  3.1(c1)}}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-5}
         & & & & $\begin{gathered}U_2>0,  U_{4} > 0,\\B_3U_1> 0 \end{gathered}$  & \rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & & $\begin{gathered} U_2<0, B_{3}H_{14}> 0\\ B_3U_1< 0\\\end{gathered}$  &\raisebox{0em}[0pt][0pt]{{Picture  3.1(c2)}} \rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & & $\begin{gathered} U_2<0, B_{3}H_{14}> 0\\ B_3U_1> 0\\\end{gathered}$  &\raisebox{-1.1em}[0pt][0pt]{{Picture  3.1(c3)}} \rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-5}
         & & & & $\begin{gathered}  U_{2} > 0,  U_{4} < 0\end{gathered}$  & \rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & & $\begin{gathered}U_2>0,  U_{4} > 0,\\ B_3U_1< 0 \end{gathered}$ &\raisebox{0em}[0pt][0pt]{\,{Picture  3.1(c4)}\,} \rule{0pt}{3.8mm}\\[0.3mm]
\cline{4-6}
         & &  & \multirow{5}*{$\begin{gathered}W_4<0\ \emph{or}\\  W_4=0\ \&\\ W_3<0 \end{gathered}$ }
         &$ B_3U_1 > 0,U_2<0$ &\raisebox{0em}[0pt][0pt]{{Picture  3.1($\stackrel{\,*}{c}$1)}}\rule{0pt}{4.4mm}\\[0.3mm]
\cline{5-6}
         & & & & $\begin{gathered} B_3U_1< 0, U_{2} < 0\end{gathered}$  &\raisebox{0em}[0pt][0pt]{{Picture  3.1($\stackrel{\,*}{c}$2)}} \rule{0pt}{4.4mm}\\[0.3mm]
\cline{5-6}
         & & & &  $\begin{gathered} B_3U_1> 0, U_{2} > 0 \end{gathered}$   &\raisebox{0em}[0pt][0pt]{\,{Picture  3.1($\stackrel{\,*}{c}$3)}\,} \rule{0pt}{4.4mm}\\[0.3mm]
\cline{5-6}
         & & & &  $\begin{gathered} B_3U_1< 0, U_{2} > 0 \end{gathered}$   &\raisebox{0em}[0pt][0pt]{\,{Picture  3.1($\stackrel{\,*}{c}$4)}\,} \rule{0pt}{4.4mm}\\[0.3mm]
 \hline
\end{tabular}
}
\end{center}
\end{table}


\begin{table}[!htb]
\begin{center}
{\scriptsize
\begin{tabular}{|c|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|}
\multicolumn{6}{c}{Table 5 (continued)}\\[1mm]
\hline $ \begin{gathered}{Configu-}\\
\emph{ration}\end{gathered}$  & $\begin{gathered}\emph{Necessary and suffi-}\\
\emph{cient conditions}
\end{gathered}$ & \multicolumn{3}{c|}{$ \begin{gathered}\emph{Additional conditions}\\ \emph{for phase
portraits}\end{gathered}$} & $\begin{gathered}\emph{Phase}\\ \emph{portrait}\end{gathered} $ \rule{0pt}{6.7mm}\\[0.4mm]
\hline
 \multirow{15}*{{Config. 3.2}} &   \multirow{15}*{$\begin{gathered}\eta>0,
                 \mu_0B_3\ne0\\ B_2=H_9= H_{13}=0\\ \emph{and either}\  \theta\ne0\ \emph{or}\\ (\theta=0\ \&\ NH_7\ne0) \end{gathered}$}
    &  \multicolumn{3}{c|}{$\begin{gathered}\mu_0<0, K<0\end{gathered}$}  &{Picture
                3.2(d1)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-6}
   &      & \multirow{6}*{$\begin{gathered}\mu_0<0,\\  K>0\end{gathered}$}& \multirow{2}*{$W_4\ge0$} &$ H_1 > 0$ &{Picture
                3.2(e1)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & &\,$ H_1 < 0\,$  &{Picture
                3.2(e2)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{4-6}
         & & &  \multirow{3}*{\,$W_4<0$\,} &$H_1>0$ &{Picture 3.2($ \stackrel{\,*}{e} $1)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & &   &\,$H_1<0,H_5<0$\, &{Picture 3.2($ \stackrel{\,*}{e} $2)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & &   &\,$H_1<0,H_5>0$\, &\,{Picture 3.2($ \stackrel{\,*}{e} $3)}\, \rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-6}
   &      & \multirow{8}*{$\begin{gathered}\mu_0>0\end{gathered}$}& \multirow{6}*{$W_4\ge0$} &\,$ B_3H_{14}<0,H_5 < 0\,$
          & {Picture 3.2(f1)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & &$\begin{gathered}B_3H_{14} < 0,H_5 > 0,   H_1 < 0\end{gathered}$ &{Picture
                3.2(f2)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & &$\begin{gathered}B_3H_{14} < 0,H_5 > 0,   H_1 > 0\end{gathered}$ &{Picture
                3.2(f3)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & &$\begin{gathered}B_3H_{14}>0,H_5<0\end{gathered}$ &{Picture
                3.2(f4)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & &$\begin{gathered}B_3H_{14}>0,H_5>0\end{gathered}$ &{Picture
                3.2(f5)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{4-6}
         & & &  \multirow{2}*{\,$W_4<0$\,} &$H_1<0$ &{Picture 3.2($ \stackrel{\,*}{f} $2)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & &   &\,$H_1>0$\, &{Picture 3.2($ \stackrel{\,*}{f} $5)}\rule{0pt}{3.8mm}\\[0.3mm]
 \hline
 \multirow{9}*{{Config. 3.3}} &   \multirow{9}*{$\begin{gathered}\eta>0,
                 \mu_0B_3\ne0\\ B_2 = H_9 = 0, H_{13}\ne0\\ \emph{and either}\  \theta\ne0\ \emph{or}\\ (\theta=0\ \&\ NH_7\ne0) \end{gathered}$}
    &  \multicolumn{3}{c|}{$\begin{gathered}\mu_0<0, K<0\end{gathered}$}  &{Picture
                3.3(d1)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-6}
   &      & \multirow{1}*{$\begin{gathered}\mu_0<0,\\  K>0\end{gathered}$}& \multicolumn{2}{c|}{$ H_5 < 0$}
          &{Picture 3.3(e1)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{4-6}
   &      &  & \multicolumn{2}{c|}{$ H_5 > 0$}
          &{Picture 3.3(e2)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-6}
   &      & \multirow{5}*{$\begin{gathered}\mu_0>0\end{gathered}$}& \multicolumn{2}{c|}{$ B_3H_{14}<0,H_5 < 0\,$}
          & {Picture 3.3(f1)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{4-6}
         & & &  \multicolumn{2}{c|}{$\begin{gathered}B_3H_{14} < 0,H_5 > 0,   H_1 < 0\end{gathered}$} &{Picture
                3.3(f2)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{4-6}
         & & & \multicolumn{2}{c|}{$\begin{gathered}B_3H_{14} < 0,H_5 > 0,   H_1 > 0\end{gathered}$} &{Picture
                3.3(f3)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{4-6}
         & & & \multicolumn{2}{c|}{$\begin{gathered}B_3H_{14}>0,H_5<0\end{gathered}$} &{Picture
                3.3(f4)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{4-6}
         & & & \multicolumn{2}{c|}{$\begin{gathered}B_3H_{14}>0,H_5>0\end{gathered}$ }&{Picture
                3.3(f5)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
 \multirow{14}*{{Config. 3.4}} &   \multirow{14}*{$\begin{gathered}\eta>0,
                 \theta B_3H_9\ne0,\\  B_2 = \mu_0=H_{14}=0 \end{gathered}$}
    &  \multirow{3}*{$\begin{gathered}  K<0\end{gathered}$} &\multirow{3}*{$W_4\ge0$}&$B_3U_1<0,U_2<0$
           &{Picture  3.4(g1)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{5-6}
    &   &  &  &$B_3U_1<0,U_2>0$  &{Picture  3.4(g2)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{5-6}
    &   &  &  &$B_3U_1 >0$  &{Picture  3.4(g3)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{4-6}
    &   &   &\multicolumn{2}{c|}{$W_4<0$}   &{Picture  3.4($ \stackrel{\,*}{g} $2)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-6}
   &  &  \multirow{10}*{$\begin{gathered}  K>0\end{gathered}$} &\multirow{4}*{$\begin{gathered}W_4>0\ \emph{or}\\  W_4=0\ \&\\ W_3\ge0 \end{gathered}$ }&$B_3U_1<0$
           &{Picture  3.4(h1)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{5-6}
    &   &  &  &$B_3U_1 > 0,U_2 < 0,H_5 < 0$  &\raisebox{-0.7em}[0pt][0pt]{{Picture  3.4(h2)}}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{5-5}
    &   &  &  &$B_3U_1>0,U_2>0$  & \rule{0pt}{3.8mm}\\[0.3mm]
 \cline{5-6}
    &   &  &  &$B_3U_1 > 0,U_2 < 0,H_5 > 0$  &{Picture  3.4(h3)}\rule{0pt}{3.8mm}\\[0.3mm]
  \cline{4-6}
   &  &  &\multirow{5}*{$\begin{gathered}W_4<0\ \emph{or}\\  W_4=0\ \&\\ W_3<0 \end{gathered}$ }&$B_3U_1<0, U_2<0$
           &{Picture  3.4($ \stackrel{\,*}{h} $1)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{5-6}
    &   &  &  &$B_3U_1>0, U_2>0$  & {Picture  3.4($ \stackrel{\,*}{h} $2)} \rule{0pt}{3.8mm}\\[0.3mm]
 \cline{5-6}
    &   &  &  &$B_3U_1>0, U_2<0$  &  {Picture  3.4($ \stackrel{\,*}{h} $3)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{5-6}
    &   &  &  &$B_3U_1<0, U_2>0$ &{Picture  3.4($ \stackrel{\,*}{h} $4)}\rule{0pt}{3.8mm}\\[0.3mm]
 \hline
%
% \multirow{6}*{{Config. 3.5}} &   \multirow{6}*{$\begin{gathered}\eta>0,
%                 \theta B_3H_9H_{14}\ne0,\\   B_2 = \mu_0=0 \end{gathered}$}
%    &  \multirow{2}*{$\begin{gathered}  K<0\end{gathered}$} & \multicolumn{2}{c|}{$ B_3U_1<0$}
%           &{Picture  3.5(g1)}\rule{0pt}{3.8mm}\\[0.3mm]
% \cline{4-6}
%    &   &  &  \multicolumn{2}{c|}{$ B_3U_1>0$} &{Picture  3.5(g2)}\rule{0pt}{3.8mm}\\[0.3mm]
% \cline{3-6}
%   &   &  \multirow{3}*{$\begin{gathered}  K>0\end{gathered}$} & \multicolumn{2}{c|}{$ B_3U_1<0$}
%           &\raisebox{-0.7em}[0pt][0pt]{{Picture  3.5(h1)}}\rule{0pt}{3.8mm}\\[0.3mm]
% \cline{4-5}
%    &   &  &  \multicolumn{2}{c|}{$ B_3U_1>0, U_3>0$} & \rule{0pt}{3.8mm}\\[0.3mm]
% \cline{4-6}
%    &   &  &  \multicolumn{2}{c|}{$ B_3U_1>0, U_3<0$} &{Picture  3.5(h2)}\rule{0pt}{3.8mm}\\[0.3mm]
% \hline
\end{tabular}
}
\end{center}
\end{table}

\begin{table}[!htb]
\begin{center}
{\footnotesize
\begin{tabular}{|c|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|}
\multicolumn{6}{c}{Table 5 (continued)}\\[1mm]
\hline $ \begin{gathered}{Configu-}\\
\emph{ration}\end{gathered} $ & $\begin{gathered}\emph{Necessary and suffi-}\\
\emph{cient conditions}
\end{gathered}$ & \multicolumn{3}{c|}{$ \begin{gathered}\emph{Additional conditions}\\ \emph{for phase
portraits}\end{gathered}$} & $\begin{gathered}\emph{Phase}\\ \emph{portrait}\end{gathered} $ \rule{0pt}{6.7mm}\\[0.4mm]
\hline
%
 \multirow{6}*{{Config. 3.5}} &   \multirow{6}*{$\begin{gathered}\eta>0,
                 \theta B_3H_9H_{14}\ne0,\\   B_2 = \mu_0=0 \end{gathered}$}
    &  \multirow{2}*{$\begin{gathered}  K<0\end{gathered}$} & \multicolumn{2}{c|}{$ B_3U_1<0$}
           &{Picture  3.5(g1)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{4-6}
    &   &  &  \multicolumn{2}{c|}{$ B_3U_1>0$} &{Picture  3.5(g2)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-6}
   &   &  \multirow{3}*{$\begin{gathered}  K>0\end{gathered}$} & \multicolumn{2}{c|}{$ B_3U_1<0$}
           &\raisebox{-0.7em}[0pt][0pt]{{Picture  3.5(h1)}}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{4-5}
    &   &  &  \multicolumn{2}{c|}{$ B_3U_1>0, U_3>0$} & \rule{0pt}{3.8mm}\\[0.3mm]
 \cline{4-6}
    &   &  &  \multicolumn{2}{c|}{$ B_3U_1>0, U_3<0$} &{Picture  3.5(h2)}\rule{0pt}{3.8mm}\\[0.3mm]
 \hline
%

 \multirow{5}*{{Config. 3.6}} &   \multirow{5}*{$\begin{gathered}\eta>0,
                 \theta B_3\ne0,\\   B_2 = \mu_0=H_{9}=0,\\ H_{13}=H_{14}=0 \end{gathered}$}
    &  \multicolumn{3}{c|}{$\begin{gathered}  K<0\end{gathered}$} &{Picture  3.6(k1)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-6}
    &   &\multirow{4}*{$\begin{gathered}  K>0\end{gathered}$}&\multicolumn{2}{c|}{$W_4\ge0$}  &{Picture  3.6(l2)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{4-6}
    &   &  &  \multirow{2}*{\ $W_4<0$\ } &\ $H_1<0$ \  &\text{ Picture  3.6($ \stackrel{\,*}{l} $ 1)}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
    &   &  &    &$H_1 >0$  &{Picture  3.6($ \stackrel{\,*}{l} $ 2)}\rule{0pt}{3.8mm}\\[0.3mm]
  \hline
 \multirow{3}*{{Config. 3.7}} &   \multirow{3}*{$\begin{gathered}\eta>0,
                 \theta B_3H_{13}\ne0,\\  B_2=\mu_0=0,\\ H_9=H_{14}=  0 \end{gathered}$}
    &  \multicolumn{3}{c|}{$\begin{gathered}  K<0\end{gathered}$} &{Picture  3.7(k1)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-6}
    &   &\multirow{2}*{$\begin{gathered}  K>0\end{gathered}$}&\multicolumn{2}{c|}{$H_5>0$}  &{Picture  3.7(l1)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{4-6}
    &   &\ &\multicolumn{2}{c|}{$H_5<0$}  &{Picture  3.7(l2)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{4-6}
  \hline
 \multirow{3}*{{Config. 3.8}} &   \multirow{3}*{$\begin{gathered}\eta>0,
                 \theta B_3 H_{14}\ne0,\\  B_2=\mu_0=   H_9=0 \end{gathered}$}
    &  \multicolumn{3}{c|}{$\begin{gathered}  K<0\end{gathered}$} &{Picture  3.8(k1)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-6}
    &   &\multirow{2}*{$\begin{gathered}  K>0\end{gathered}$}&\multicolumn{2}{c|}{$H_5<0$}  &{Picture  3.8(l1)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{4-6}
    &   &\ &\multicolumn{2}{c|}{$H_5>0$}  &{Picture  3.8(l2)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{4-6}
\hline
   \multirow{14}*{{Config. 3.9}} &
   \multirow{14}*{$\begin{gathered}\eta=0, B_2=0,\\
                 M\theta\mu_0B_3H_9\ne0  \end{gathered}$}& \multirow{7}*{$\begin{gathered}\mu_0<0\end{gathered}$}& \multirow{3}*{$\begin{gathered}W_4>0\ \emph{or}\\  W_4=0\ \&\\ W_3\ge0 \end{gathered}$ }
         &$ U_4 < 0, H_4<0$ &\raisebox{0em}[0pt][0pt]{{Picture  3.9(b1)}}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & & $\begin{gathered}  U_4 < 0, H_4> 0\end{gathered}$  & {{Picture  3.9(b2)}} \rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & & $\begin{gathered}  U_4 > 0\end{gathered}$  & {{Picture  3.9(b3)}} \rule{0pt}{3.8mm}\\[0.3mm]
\cline{4-6}
         & &  & \multirow{3}*{$\begin{gathered}W_4<0\ \emph{or}\\  W_4=0\ \&\\ W_3<0 \end{gathered}$ }
         &$ B_3U_1 > 0, U_4<0$ &\raisebox{0em}[0pt][0pt]{{Picture  3.9($\stackrel{*}{b}$2)}}\rule{0pt}{4.8mm}\\[0.3mm]
\cline{5-6}
         & & & & $\begin{gathered}   B_3U_1 > 0, U_4>0 \end{gathered}$  &{Picture  3.9($\stackrel{*}{b}$3)} \rule{0pt}{4.8mm}\\[0.3mm]
\cline{5-6}
         & & & & $\begin{gathered} B_3U_1 < 0\end{gathered}$  &\raisebox{0em}[0pt][0pt]{{Picture  3.9($\stackrel{*}{b}$4)}} \rule{0pt}{4.8mm}\\[0.3mm]
\cline{3-6}
         & & \multirow{7}*{$\begin{gathered}\mu_0>0\end{gathered}$}& \multirow{5}*{$\begin{gathered}W_4\ge0 \end{gathered}$ }
         &$  B_3U_1 < 0, U_4<0$ &\raisebox{-0.7em}[0pt][0pt]{{Picture  3.9(c1)}}\rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-5}
         & & & & $\begin{gathered}B_3U_1 > 0, H_4>0\end{gathered}$  & \rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & & $\begin{gathered} B_3U_1 < 0, U_4>0 \end{gathered}$  &\raisebox{0em}[0pt][0pt]{{Picture  3.9(c2)}} \rule{0pt}{3.8mm}\\[0.3mm]
\cline{5-6}
         & & & & $\begin{gathered} B_3U_1 > 0, H_4<0 \end{gathered}$  &\raisebox{0em}[0pt][0pt]{{Picture  3.9(c3)}} \rule{0pt}{3.8mm}\\[0.3mm]
\cline{4-6}
         & &  & \multirow{2}*{$\begin{gathered}W_4<0\end{gathered}$ }
         &$ B_3U_1 > 0 $ &\raisebox{0em}[0pt][0pt]{{Picture  3.9($\stackrel{\,*}{c}$1)}}\rule{0pt}{4.4mm}\\[0.3mm]
\cline{5-6}
         & & & & $\begin{gathered} B_3U_1< 0 \end{gathered}$  &\raisebox{0em}[0pt][0pt]{{Picture  3.9($\stackrel{\,*}{c}$2)}} \rule{0pt}{4.4mm}\\[0.3mm]
\hline
 \multirow{3}*{{Config. 3.10}} &
 \multirow{3}*{$\begin{gathered}\eta=0,  B_2=H_9=0,\\  M\theta B_3\mu_0H_{13}\ne0    \end{gathered}$}
    &  \multicolumn{3}{c|}{$\begin{gathered}  \mu_0<0\end{gathered}$} &{Picture  3.10(e1)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-6}
    &   &\multirow{2}*{$\begin{gathered}  \mu_0>0\end{gathered}$}&\multicolumn{2}{c|}{$B_3U_1<0$}  &{Picture  3.10(f1)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{4-6}
    &   &\ &\multicolumn{2}{c|}{$B_3U_1>0$}  &{Picture  3.10(f2)}\rule{0pt}{3.8mm}\\[0.3mm]
 \hline
 \multirow{4}*{{Config. 3.11}} &
 \multirow{4}*{$ \begin{gathered}\eta=0,  M\theta B_3\mu_0 \ne 0,\\ B_2=H_9=H_{13}=0 \\     \end{gathered} $}
    &  \multicolumn{3}{c|}{$\begin{gathered}  \mu_0<0\end{gathered}$} &{Picture  3.11($\stackrel{\,*}{e}$1)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-6}
    &   &\multirow{4}*{$\begin{gathered}  \mu_0>0\end{gathered}$}&\multirow{2}*{$W_4\ge0$} &{$\begin{gathered}H_5>0 \end{gathered}$}  &{Picture  3.11(f1)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{5-6}
    &   & &  &{$H_5<0$}  &{Picture  3.11(f2)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{4-6}
    &   &\ &\multicolumn{2}{c|}{$W_4<0$}  &{Picture  3.11($\stackrel{\,*}{f}$1)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
% \multirow{4}*{{Config. 3.12}} &
% \multirow{4}*{$\begin{gathered}\eta=0, M\theta B_3 H_9\ne0,\\ B_2=\mu_0=0 \\     \end{gathered}$}
%    & \multirow{2}*{$\begin{gathered}W_4>0\end{gathered}$} & \multicolumn{2}{c|}{$H_5<0$} &{Picture  3.12(h3)}\rule{0pt}{3.8mm}\\[0.3mm]
% \cline{4-6}
%    &   & & \multicolumn{2}{c|}{$H_5>0$} &{Picture  3.12(h2)}\rule{0pt}{3.8mm}\\[0.3mm]
% \cline{3-6}
%    &   &\multirow{2}*{$\begin{gathered}W_4<0\end{gathered}$} &  \multicolumn{2}{c|}{$B_3U_1>0$}  &{Picture  3.12($\stackrel{\,*}{h}$2)}\rule{0pt}{3.8mm}\\[0.3mm]
% \cline{4-6}
%    &   &   &\multicolumn{2}{c|}{$B_3U_1<0$}   &{Picture  3.12($\stackrel{\,*}{h}$1)}\rule{0pt}{3.8mm}\\[0.3mm]
%\hline
% {{Config. 3.13}} &    {$\begin{gathered}\eta=0,
%                  M\theta B_3 \ne0,\\ B_2=\mu_0=H_9=0 \\     \end{gathered}$}
%    & \multicolumn{3}{c|}{$-$} &{Picture  3.13($\stackrel{\,*}{l}$ 1)}\rule{0pt}{3.8mm}\\[0.3mm]
%  \hline
\end{tabular}
}\end{center}
\end{table}

% created at 12 M



\begin{table}[!htb]
\begin{center}
{\footnotesize
\begin{tabular}{|c|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|}
\multicolumn{6}{c}{Table 5 (continued)}\\[1mm]
\hline $ \begin{gathered}{Configu-}\\
\emph{ration}\end{gathered} $ & $\begin{gathered}\emph{Necessary and suffi-}\\
\emph{cient conditions}
\end{gathered}$ & \multicolumn{3}{c|}{$ \begin{gathered}\emph{Additional conditions}\\ \emph{for phase
portraits}\end{gathered}$} & $\begin{gathered}\emph{Phase}\\ \emph{portrait}\end{gathered} $ \rule{0pt}{6.7mm}\\[0.4mm]
\hline
%
 \multirow{4}*{{Config. 3.12}} &
 \multirow{4}*{$\begin{gathered}\eta=0, M\theta B_3 H_9\ne0,\\ B_2=\mu_0=0 \\     \end{gathered}$}
    & \multirow{2}*{$\begin{gathered}W_4>0\end{gathered}$} & \multicolumn{2}{c|}{$H_5<0$} &{Picture  3.12(h3)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{4-6}
    &   & & \multicolumn{2}{c|}{$H_5>0$} &{Picture  3.12(h2)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-6}
    &   &\multirow{2}*{$\begin{gathered}W_4<0\end{gathered}$} &  \multicolumn{2}{c|}{$B_3U_1>0$}  &{Picture  3.12($\stackrel{\,*}{h}$2)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{4-6}
    &   &   &\multicolumn{2}{c|}{$B_3U_1<0$}   &{Picture  3.12($\stackrel{\,*}{h}$1)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
 {{Config. 3.13}} &    {$\begin{gathered}\eta=0,
                  M\theta B_3 \ne0,\\ B_2=\mu_0=H_9=0 \\     \end{gathered}$}
    & \multicolumn{3}{c|}{$-$} &{Picture  3.13($\stackrel{\,*}{l}$ 1)}\rule{0pt}{3.8mm}\\[0.3mm]

  \hline
\end{tabular}
}\end{center}
\end{table}



\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.90\textwidth]{fig5} % Portraits-LV-3p1-3p7
\end{center}
 \caption{Phase portraits
of the family of LV-systems with exactly three invariant
lines}\label{Fig:Ph-Port-3ILs}
\end{figure}


\addtocounter{figure}{-1}
\renewcommand{\thefigure}{\arabic{figure} (cont.)}
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.90\textwidth]{fig5cont} % Portraits-LV-3p8-3p13
\end{center}
 \caption{Phase portraits
of the family of LV-systems with exactly three invariant lines}
\end{figure}
\renewcommand{\thefigure}{\arabic{figure}}


\begin{remark}\label{rem:change-sys-3.1} \rm
Observe that  substituting
$(x,y,t,f,g,h)$ by $(fy,fx,t/f,1/f,h,g)$ keeps systems
\eqref{sys:Config.3.1-G} and interchanges
 the two invariant lines  and therefore also interchanges the
corresponding  singularities.
\end{remark}

Considering  \eqref{expr:Delta-i,delta-i:3.1} the condition
\eqref{cond:A1} is equivalent to the condition
\begin{equation}\label{cond:A1-Delta-i}
\Delta_1 \Delta_2 \Delta_3\Delta_4
\widetilde{\Delta}_1\widetilde{\Delta}_2\widetilde{\Delta}_3
(\widetilde{\Delta}_1+\widetilde{\Delta}_2)
(\widetilde{\Delta}_1+\widetilde{\Delta}_3)\delta_1 \rho_4\ne0
\end{equation}

Taking into account  \eqref{expr:Delta-i,delta-i:3.1} and
\eqref{ISPs:tilde delta-i:3.1} we evaluate for systems
\eqref{sys:Config.3.1-G} the  invariant polynomials we need:
\begin{equation}\label{val:inv-Conf-3.1}
\begin{gathered}
\mu_0= g h (g + h-1)=-\widetilde{\Delta}_1\widetilde{\Delta}_2\widetilde{\Delta}_3,\\
     K= 2 g(g-1) x^2 + 4 g h x y + 2 h(h-1)y^2,\\
W_4 = \mu_0^2\,\delta_1\delta_2\delta_3\delta_4,\quad W_3=
\mu_0^2
\sum_{1\le i<j<l\le 4}    \delta_i\,\delta_j\,\delta_l,\\
H_9= -576 f^2 (1 - g + f g)^2 (f+ h - f h)^2=-576{\Delta}_2^2 {\Delta}_3^2,\\
 H_{14}= 30(1-f)g h (f g + h)
 =30\rho_4 \widetilde{\Delta}_1\widetilde{\Delta}_2 \widetilde{\Delta}_3(1-f),\\
 B_3= 3 ( 1- f) (f g + h) x^2 y^2=3\widetilde{\Delta}_1\rho_4(1-f)x^2y^2,\\
 U_{1}= \frac{1}{8}f(1-f)(g-1)^2(h-1)^2(f g + h)=
          \frac{1}{8}\Delta_1\widetilde{\Delta}_1\rho_4  (1 - f) (g-1)^2 (h-1)^2 ,\\
 U_{2}=  3 f (g-1 - f g) (fh-f - h)= 3\Delta_2\Delta_3\widetilde{\Delta}_2
 \widetilde{\Delta}_3,\\
 U_{3}=  \frac{1}{2}(1 + f) (g-1) (h-1) (f g + h)=
   \frac{1}{2}\widetilde{\Delta}_1\rho_1\rho_4(g-1) (h-1),\\
U_{4}= f (g-1 - f g) ( f h-f - h )( f g -2 f- f^2 g - h + f h).
\end{gathered}
\end{equation}
Denoting $\mathcal{A}=f (g-1 - f g)= {\Delta}_1
{\Delta}_2\widetilde{\Delta}_2$ and
$\mathcal{B}=(fh-f - h)={\Delta}_3\widetilde{\Delta}_3/{\Delta}_1$ the invariants $U_2$
and $U_4$   could be represented  in the following forms,
respectively:
$$
 U_2=3\mathcal{A}\mathcal{B},\quad U_4=\mathcal{A}\mathcal{B}(\mathcal{A}+\mathcal{B}).
$$

\begin{remark}\label{rem:signs-U2,U4,B3U1,B3-nu} \rm
 We note that provided the condition
\eqref{cond:A1} is satisfied, by \eqref{val:inv-Conf-3.1}the
following relations hold:
\begin{itemize}
  \item $B_3U_1=\frac{3}{8}\Delta_1\rho_4^2\widetilde{\Delta}_1^{2}
  (g-1)^2 (h-1)^2 (f-1)^2x^2y^2$
$\Rightarrow$ $\operatorname{sign}(B_3U_1)= \operatorname{sign}(\Delta_1)$;

 \item  $ \operatorname{sign}(U_2)=  \operatorname{sign}(\Delta_2\Delta_3\widetilde{\Delta}_2
 \widetilde{\Delta}_3)$;

  \item If  $U_2>0$ \quad $\Rightarrow $\quad $\operatorname{sign}(U_4)= \operatorname{sign}(\Delta_1\Delta _2\widetilde{\Delta}_2) = \operatorname{sign}(\Delta_1\Delta
 _3\widetilde{\Delta}_3)$;

  \item $B_3H_{14}=90\rho_4^2 \widetilde{\Delta}_2
\widetilde{\Delta}(f-1)^2\widetilde{\Delta}_1^2x^2y^2\quad
\Rightarrow\quad \operatorname{sign}(B_3H_{14})= \operatorname{sign}(\widetilde{\Delta}_2
\widetilde{\Delta}_3)$.
 \end{itemize}
 \end{remark}

\paragraph{The case $\mu_0<0$.}
 As $\mu_0=\operatorname{Discrim}(K)/16$ we conclude that $K(x,y)$ ie a binary form
with well defined sign and we   shall consider two subcases: $K<0$
and $K>0$.

\subparagraph{The subcase $K<0$.} Then according to
\cite{ArtLliVul08-IJBCh} (see Table 1) on the finite part of the
phase plane, systems \eqref{sys:Config.3.1-G} possess three
saddles and one anti-saddle. Moreover the anti-saddle is a node if
$W_4\ge0$ and it is of the center-focus type if $W_4<0$.   In the
second case, by Lemma \ref{lem:M4-center} we have a strong focus.

Since at infinity there exist three real distinct singularities,
according to the index theory  all of them must be nodes.

 {\it I. Assume first $W_4\ge0$.} We claim that in
this case the phase portraits of systems \eqref{sys:Config.3.1-G}
correspond to one of those indicated below if and only if the
conditions indicated on the right are  satisfied:
\begin{gather*}
 \text{Picture 3.1(a1)} \Leftrightarrow   B_3U_1 < 0,\, U_{2} < 0; \\
 \text{Picture 3.1(a2)} \Leftrightarrow    B_3U_1 < 0,\, U_{2}> 0; \\
 \text{Picture 3.1(a3)} \Leftrightarrow    B_3U_1 > 0.
 \end{gather*}
Indeed, since  by Remark \ref{rem:signs-U2,U4,B3U1,B3-nu} we have
$\operatorname{sign}(\Delta_1)=\operatorname{sign}(B_3U_1)$ we need to  consider two cases.

(1) If  $B_3U_1 < 0$ then $\Delta_1<0$ and hence the
singular point $M_1(0,0)$ is a saddle.  Since all infinite
singular points are nodes this implies $\widetilde{\Delta}_i>0$,
$i=1,2,3$. By Remark
 \ref{rem:signs-U2,U4,B3U1,B3-nu} we get $\operatorname{sign}(U_2)=\operatorname{sign}(\Delta_2\Delta_3)$.

Thus, if $U_2<0$ then $\Delta_2\Delta_3<0$ and one of the points
$M_2$ or $M_3$ is a node and the remaining points are saddles.
This univocally leads to {Picture 3.1(a1)}.

Assuming $U_2>0$ we obtain  $\Delta_2\Delta_3>0$, i.e. both points
are saddles and the point $M_4$ is a node. In this case we clearly
have {Picture 3.1(a2)}.

(2) Suppose  $B_3U_1 > 0$. Then $\Delta_1>0$ and hence the
singular point $M_1(0,0)$ is a node, whereas the remaining tree
finite singularities are saddles. As the infinite singular points
are nodes we get {Picture 3.1(a3)}.  This completes the
proof of our claim.


 {\it II. Assume  $W_4<0$.}  Then on the phase plane, apart from the three
saddles  there exists a focus which clearly could only be the
singular point $M_4$. It is known that this point must be located
inside the triangle formed by other three saddle points (see for
instance \cite{Berl}).

We claim that there  could not be a separatrix connecting $M_2$
 with $M_3$. Indeed, suppose   that such
a connection exists.  This connection could not belong to an
invariant line, otherwise systems \eqref{sys:Config.3.1-G} possess
four invariant lines and we get a contradiction. Suppose that this
connection is different from the segment $M_2M_3$ of a line.  Then
we obtain a closed domain bordered by this segment and the
separatrix connecting $M_2$ and $M_3$ (at which we have two
saddles) and clearly on the segment of the straight line $M_2M_3$
there must be at least one point  of contact. Therefore according
to \cite[Theorem 2.5]{Chicone-Tian} the straight line passing
through these singular points must be an invariant line and we
again get the contradiction mentioned above. These arguments lead
univocally to a  phase portrait which is topologically equivalent
to {Picture 3.1($ \stackrel{*}{a} $2)}.

\subparagraph{The subcase $K>0$} Then according to
\cite{ArtLliVul08-IJBCh} (see Table 1) systems
\eqref{sys:Config.3.1-G} possess one saddle and  three
anti-saddles. Clearly only one anti-saddle could be a focus and
considering \cite{ArtLliVul08-IJBCh} (see Table 1) besides the
saddle we have three nodes if either $W_4>0$ or $W_4=0$ and
$W_3\ge0$; and we have two nodes and a focus  if either $W_4<0$ or
$W_4=0$ and $W_3<0$.

 {\it I. Assume first  $W_4>0$ or
 $W_4=0$ and $W_3\ge0$.} We claim that in this case the phase
portraits of   systems \eqref{sys:Config.3.1-G}  correspond to one
of those  indicated below if and only if the   conditions
indicated on the right are  respectively satisfied:
\begin{gather*}
 \text{Picture 3.1(b1)} \Leftrightarrow
             \text{either $B_3U_1 < 0$, or $B_3U_1 > 0$, $U_2>0$,
	      $U_4>0$, $U_3>0$};   \\
\begin{aligned}
 \text{Picture 3.1(b2)} \Leftrightarrow &   B_3U_1 > 0 \text{ and either }
              U_2<0, B_3H_{14}>0,\\
   &\text{or } U_2>0, U_4>0, U_3<0;
\end{aligned} \\
 \text{Picture 3.1(b3)} \Leftrightarrow  B_3U_1 > 0 \text{ and  either }
  U_2<0, B_3H_{14}<0,
  \text{ or } U_2>0, U_4<0.
\end{gather*}

Indeed, first of all we observe that at infinity we must have one
node and two saddles (the sum of the indexes must be -1). Then due
to Remark \ref{rem:change-sys-3.1} without loss of generality we
may assume that $R_2(1,0,0)$ is a saddle, i.e.
$\widetilde{\Delta}_2<0$.

As it was mentioned previously,  the type of the singular point
$M_1(0,0)$ depends on the sign of the  invariant polynomial
$B_3U_1$. So we   consider   two cases.

(1) If  $B_3U_1 < 0$ then $\Delta_1<0$ and hence the
singular point $M_1(0,0)$ is a saddle  and the other three
singularities   are nodes. As   $R_2(1,0,0)$ is a saddle,  by
\ref{ISPs:tilde delta-i:3.1} we have $g<0$ and this fixes the
position of the finite node $M_2(-1/g,0)$. We claim   that in this
case the singular point $R_3(0,1,0)$ could not be a saddle.
Indeed, assuming the contrary we have $\widetilde{\Delta}_3<0$
(i.e. $h<0$).  Since $f<0$,    the domain in the second quadrant
bordered by the three invariant lines (one being the line at
infinity) has on its border exactly  three  singularities, which
are saddles. Moreover the singular point $M_4$ inside this domain
is forced in this case to be a focus or a center. So we obtain a
contradiction which proves our claim.

Thus $R_3(0,1,0)$ is a node (i.e. $h>0$) and we get univocally a
phase portrait topologically equivalent to {Picture
3.1(b1)}.

(2) Assume now   $B_3U_1 > 0$, i.e. the singular point
$M_1(0,0)$ is a node. As by our assumption $R_2(1,0,0)$ is a
saddle (i.e. $\widetilde{\Delta}_2<0$), considering Remark
\ref{rem:signs-U2,U4,B3U1,B3-nu} we obtain $\operatorname{sign}(U_2)= -\operatorname{sign}(
\Delta_2\Delta_3 \widetilde{\Delta}_3)$.

(a)  If   $U_2<0$ then  we obtain $\Delta_2\Delta_3
\widetilde{\Delta}_3>0$.  On the other hand by Remark
\ref{rem:signs-U2,U4,B3U1,B3-nu} we have $\operatorname{sign}(B_3H_{14})=
\operatorname{sign}(\widetilde{\Delta}_2 \widetilde{\Delta}_3)$ and
 we shall consider two subcases: $B_3H_{14}>0$ and $B_3H_{14}<0$.

 ($\alpha$)   Suppose first $B_3H_{14}>0$. Since
$\widetilde{\Delta}_2<0$ we obtain $\widetilde{\Delta}_3<0$, i.e.
$h<0$.  Therefore the condition $U_2<0$ implies
$\Delta_2\Delta_3<0$, i.e. one of the singularities $M_2(-1/g,0)$
or $M_3(0,-f/h)$ is a saddle and the other   is a node. As both
infinite singular points $R_2(1,0,0)$ and $R_3(0,1,0)$ are of the
same types (saddles), due to Remark \ref{rem:change-sys-3.1}
without loss of generality we may assume that $M_2(-1/g,0)$ is a
saddle. Taking into account that the finite saddle must be inside
the triangle formed by three finite nodes we obviously get a phase
portrait topologically equivalent to   {Picture 3.1(b2)}.

($\beta$) In the case $B_3H_{14}<0$ we obtain
$\widetilde{\Delta}_3>0$, i.e. $h>0$ and hence, besides the saddle
$R_2$ at infinity we have the node $R_3$ and the saddle $R_1$. In
this case considering the condition $U_2<0$ we get
$\Delta_2\Delta_3>0$ and this means that both  singular points
$M_2$ and $M_3$ are nodes. Clearly the remaining point $M_4$ is a
saddle. Then considering the location of the singularities we get
a phase portrait topologically equivalent to {Picture
3.1(b3)}.

(b) Suppose now $U_2>0$.  Then due to the Remark
\ref{rem:signs-U2,U4,B3U1,B3-nu}   and $\widetilde{\Delta}_2<0$ we
get $\Delta_2\Delta_3 \widetilde{\Delta}_3<0$. Moreover, since
$U_2>0$ we have $\operatorname{sign}(U_4)= \operatorname{sign}(\Delta_1\Delta
_3\widetilde{\Delta}_3)$ and since $ \Delta_1>0$ then clearly we
obtain $\operatorname{sign}(U_4)= \operatorname{sign}(\Delta _3\widetilde{\Delta}_3)=-
\operatorname{sign}(\Delta_2$).

 ($\alpha$)  Admit  first $U_4>0$,
 i.e. $\Delta _3\widetilde{\Delta}_3>0$. So, we have $\Delta_2<0$ (i.e.
$M_2$ is a saddle) and then  $M_3$ and $M_4$ are nodes. Hence
$\Delta _3>0$ and this implies $\widetilde{\Delta}_3>0$, i.e.
$R_3$ is a node.  Taking into consideration  the node $R_3$ and
the saddles $R_2$ and $R_1$ at infinity, we arrive at the two
possibilities given by the   \emph{Portraits }\emph{A} and
\emph{B}. We observe that for the first (respectively second)
phase portraits the nodes $M_1$ and $M_4$ have different
(respectively the same) stability. As $\rho_1=1+f>0$ we conclude
that the realization of each portrait depends on the sign of
$\rho_4$. We point out that the singularity $M_4$ changes its
stability when $\rho_4$ change the sign (passing through zero) and
since $\rho_4\ne0$ (due to $B_3\ne0$)   we could not have  a
separatrix connection $M_2R_1$.


 \noindent
\begin{minipage}{0.59\textwidth}\label{Figures-1-2}
 On the other hand, as $R_1$ is a saddle (i.e. $1-g-h~<~0$)
considering $g<0$ we get  $h>1-g>1$. Therefore according to
\eqref{val:inv-Conf-3.1} we obtain $\operatorname{sign}(U_3)=
\operatorname{sign}(\rho_1\rho_4)$. Thus, we get {Portrait A} if $U_3<0$
and {Portrait  B} if $U_3>0$. It remains to note, that  the
phase portrait given by {Portrait A} (respectively by
{Portrait B}) is topologically equivalent to
{Picture 3.1(b2)} (respectively   {Picture
3.1(b1)}).
\end{minipage}\quad
\begin{minipage}{0.35\textwidth}
\centerline{\includegraphics[width=2.30cm]{figure1a}\quad
\includegraphics[width=2.30cm]{figure2a}}
\centerline{\quad Portrait  A \hspace{10mm} Portrait B\quad }
\end{minipage}


 ($\beta$)  Assuming  $U_4<0$ we have $\Delta_2>0$ (i.e. $M_2$ is
a node). So we have two possibilities: either $M_3$ is a saddle
and  $M_4$ is a node, or $M_3$ is a node and $M_4$ is a saddle. In
the first (respectively the second)  case due to $\Delta
_3\widetilde{\Delta}_3<0$ we obtain that $R_3$   is a node
(respectively a saddle) and hence $R_1$ is a saddle (respectively
a node). So in both   cases we get  phase portraits which are
topologically equivalent to {Picture~3.1(b3)}.

 {\it II. Suppose now  $W_4<0$ or  $W_4=0$ and $W_3<0$.}
It was mentioned earlier that systems \eqref{sys:Config.3.1-G}
possess as finite singularities a saddle, two nodes and a focus.
We claim that in this case the phase portraits of these  systems
correspond  to one of the portraits indicated below if and only if
the  conditions given   on the right are respectively satisfied:
\begin{gather*}
 \text{Picture 3.1($\stackrel{*}{b}$1)} \Leftrightarrow
             \text{either } B_3U_1 < 0, U_{2} < 0,
 \text{ or }    B_3U_1 > 0, U_2>0, U_4>0, U_3>0;   \\
\text{Picture 3.1($\stackrel{*}{b}$2)} \Leftrightarrow  B_3U_1
>0  \text{ and either }   U_{2} < 0,
 \text{ or }     U_2>0, U_4>0, U_3<0;  \\
\text{Picture 3.1($\stackrel{*}{b}$3)} \Leftrightarrow
 B_3U_1 > 0, U_{2} > 0, U_4<0; \\
\text{Picture 3.1($\stackrel{*}{b}$4)} \Leftrightarrow
 B_3U_1 < 0, U_{2} > 0.
 \end{gather*}
Indeed, first of all we recall  that the singular point of the
focus type could be only $M_4$.

(1) \emph{The subcase  $B_3U_1 < 0$.}  Then by Remark
\ref{rem:signs-U2,U4,B3U1,B3-nu} we have $\Delta_1<0$ (yielding
$f<0$), i.e. $M_1(0,0)$ is a saddle and $M_2$ and $M_3$ are nodes
($M_4$ being a focus). At infinity we have the same singularities:
two saddles and one node. Thus similarly to the previous case, due
to Remark \ref{rem:change-sys-3.1} we may assume that $R_2(1,0,0)$
is a saddle. So   one of the remaining infinite points $R_3$ and
$R_1$ is a saddle and other one is a node. Moreover, since
$\Delta_2>0$ and $\Delta_3>0$, by Remark
\ref{rem:signs-U2,U4,B3U1,B3-nu} we find out that
$\operatorname{sign}(U_2)=-\operatorname{sign}(\widetilde{\Delta}_3)$. So we shall examine two
cases: $U_2<0$ and $U_2>0$.

(a)  If   $U_2<0$ then  $\widetilde{\Delta}_3>0$   and hence
$R_3$ is node and $R_1$ is a saddle.  Considering the focus
$M_4(x_4,y_4)$   we univocally arrive  to the
{Picture~3.1($\stackrel{*}{b}$1)}.

(b)    Admit now $U_2>0$.  Then $\widetilde{\Delta}_3<0$ and
therefore the singular point $R_3$ is a saddle and   $R_1$ is a
node. Taking into account the location of all the singularities in
this case we get {Picture 3.1($\stackrel{*}{b}$4)}.


(2) \emph{The subcase  $B_3U_1 > 0$.} Considering Remark
\ref{rem:signs-U2,U4,B3U1,B3-nu} we have   $\Delta_1>0$ (i.e.
$f>0$) and therefore $M_1(0,0)$ is a node. Hence
$\Delta_2\Delta_3<0$ (since $M_4$ is a focus) and as
$\tilde\Delta_2<0$ according to Remark
\ref{rem:signs-U2,U4,B3U1,B3-nu} we obtain $\operatorname{sign}(U_{2})=
\operatorname{sign}(\widetilde{\Delta}_3$).

(a)   Suppose $U_2<0$.  Then $\widetilde{\Delta}_3<0$   and hence
$R_3(0,1,0$) is a saddle. Therefore since the infinite singularity
$R_2(1,0,0)$ is also a saddle, then without loss of generality we
may assume that the point $M_2(-1/g,0)$ is a saddle due to Remark
\ref{rem:change-sys-3.1}. Then   considering the positions of the
focus $M_4$ and of the node $R_1$ we  univocally arrive to the
{Picture 3.1($\stackrel{*}{b}$2)}.

(b)  Assume now $U_2>0$. Then we get $\widetilde{\Delta}_3>0$
(i.e. $h>0$) and in this case the singular point $R_3$ is a node
and the third infinite point $R_1$ is a saddle. Hence according to
Remark \ref{rem:signs-U2,U4,B3U1,B3-nu} we have $ \operatorname{sign}(U_4)=
\operatorname{sign}(\Delta_1\Delta _3\widetilde{\Delta}_3)= \operatorname{sign}(\Delta_3)$.

 ($\boldsymbol{\alpha}$) Admit first $U_4<0$. Then $\Delta_3<0$ (i.e. $M_3$ is
a saddle) and  therefore  $M_2$ is a node. Taking into
consideration the location of all the singularities of a system
\eqref{sys:Config.3.1-G} in the case under consideration we obtain
univocally   the phase portrait given by
{Picture 3.1($\stackrel{*}{b}$3)}.

($\boldsymbol{\beta}$) If  $U_4>0$ then $\Delta_3>0$, i.e. $M_3$ is
a node  and hence $M_2$ is a saddle. So, in the same manner as in
the   case   with one saddle and three nodes, we obtain two
different phase portraits given by \emph{Portrait A} and
\emph{Portrait B} (see page \pageref{Figures-1-2}) but with
$M_4$  as focus   instead of a node. Moreover, it is clear, that
the stabilities of the node $M_1$ and of the focus $M_4$
distinguish these two phase portraits. More exactly, we obtain
\emph{Portrait A} if $\rho_1\rho_4<0$ and \emph{Portrait B} if
$\rho_1\rho_4>0$. It remains to remark, that \emph{Portrait A}
(respectively \emph{Portrait B}) is topologically equivalent to
{Picture 3.1($\stackrel{*}{b}$2)} (respectively
{Picture 3.1($\stackrel{*}{b}$1)}) and according to
\eqref{val:inv-Conf-3.1}, in this case we have
$\operatorname{sign}(\rho_1\rho_4)=\operatorname{sign}(U_3)$.

\paragraph{The case $\mu_0>0$.} Then according to \cite {ArtLliVul08-IJBCh}
(see Table 1) for systems \eqref{sys:Config.3.1-G}, on the finite part of the
phase plane  there are two saddles and two anti-saddles. Moreover
as for systems in this family one anti-saddle is always  a node,
by \cite{ArtLliVul08-IJBCh} we conclude that both anti-saddles are
nodes if either $W_4>0$ or $W_4=0$ and $W_3\ge0$; and   one of
them is a focus     if either $W_4<0$ or $W_4=0$ and $W_3<0$.

On the other hand since the three singular points at infinity are
simple, then by the index theory we must have two nodes and a
saddle. So due to Remark \ref{rem:change-sys-3.1} we may assume
that the point $R_2(1,0,0)$ is a node, i.e. $\tilde \Delta_2>0$
(this yields $g>0$).

\subparagraph{\it Assume first  $W_4>0$ or  $W_4=0$ and
$W_3\ge0$.} So systems \eqref{sys:Config.3.1-G} have two saddles
and two nodes. We claim that the phase portrait of a system in
this family is given by one of the portraits indicated below if
and only if the   conditions on the right are respectively
satisfied:
\begin{gather*}
 \text{Picture 3.1(c1)} \Leftrightarrow
 \text{ either }  U_2<0, B_3H_{14}<0,
 \text{ or } U_2>0, U_4>0, B_3U_1>0; \\
\text{Picture 3.1(c2)} \Leftrightarrow
   U_2<0,  B_3H_{14}>0,  B_3U_1<0;  \\
 \text{Picture 3.1(c3)} \Leftrightarrow
 \text{ either }  U_2<0, B_3H_{14}>0, \,  B_3U_1>0, \\
   \text{ or }   U_2>0, U_4<0;   \\
\text{Picture 3.1(c4)} \Leftrightarrow
  U_2>0, U_4>0,   B_3U_1<0.
\end{gather*}
Indeed first  we recall that the type of the singular point
$M_1(0,0)$ is governed by the invariant polynomial $B_3U_1$.

 {\it I. The subcase  $B_3U_1 < 0$.}  Then by Remark
\ref{rem:signs-U2,U4,B3U1,B3-nu} it follows $\Delta_1<0$ (yielding
$f<0$),  i.e. $M_1(0,0)$ is a saddle.

(1)  {\it Suppose first $U_2<0$.}  Then by Remark
\ref{rem:signs-U2,U4,B3U1,B3-nu} we get ${\Delta}_2{\Delta}_3
\widetilde{\Delta}_2\widetilde{\Delta}_3<0$ and $\operatorname{sign}(B_3H_{14})=
\operatorname{sign}(\widetilde{\Delta}_2 \widetilde{\Delta}_3)$.

(a) If $B_3H_{14}<0$ then
$\widetilde{\Delta}_2\widetilde{\Delta}_3<0$ and this implies
${\Delta}_2{\Delta}_3>0$. Therefore (as $M_1$ is a saddle) both
points $M_2$ and $M_3$ are nodes. On the other hand since
$\widetilde{\Delta}_2>0$ we obtain $\widetilde{\Delta}_3<0$, i.e.
$R_3$ is a saddle.  So taking into considerations the locations of
the finite singularities   we arrive univocally at a phase
portrait  topologically equivalent to {Picture 3.1(c1)}.

(b) In the case $B_3H_{14}>0$ we obtain
$\widetilde{\Delta}_2\widetilde{\Delta}_3>0$ (i.e. $R_2$ and $R_3$
are both nodes)  and this implies ${\Delta}_2{\Delta}_3<0$.
Therefore one of the points   $M_2$ or $M_3$ is a saddle  and due
to Remark \ref{rem:change-sys-3.1}, without loss of generality we
may assume that such a saddle is $M_2$. So in this case   we get
univocally the {Picture 3.1(c2)}.


(2) {\it Assume now  $U_2>0$.}  Then ${\Delta}_2{\Delta}_3
\widetilde{\Delta}_2\widetilde{\Delta}_3>0$ and according to
Remark \ref{rem:signs-U2,U4,B3U1,B3-nu}  due  to the condition
$\Delta_1<0$ we obtain $ \operatorname{sign}(U_4)= -\operatorname{sign}(\Delta
_2\widetilde{\Delta}_2) =-\operatorname{sign}(\Delta _3\widetilde{\Delta}_3). $

(a) If $U_4<0$ then we obtain $\Delta _2\widetilde{\Delta}_2>0$
and $\Delta _3\widetilde{\Delta}_3>0$. Since by assumption
$\widetilde{\Delta}_2>0$ we obtain $\Delta _2>0$, i.e. the
singular point $M_2$ is a node. We claim that in this case the
singular point $R_3$ could not be a saddle. Indeed, supposing the
contrary, we obtain $\widetilde{\Delta}_3=h<0$ and since $f<0$
this implies ${\Delta}_3=f(fh-f-g)/h>0$. However  this contradicts
$\Delta _3\widetilde{\Delta}_3>0$ and our claim is proved.

Thus   we get the conditions $\Delta_2>0$, $\Delta_3>0$,
$\widetilde{\Delta}_2>0$ and $\widetilde{\Delta}_3>0$. In other
words all the singularities $M_2$, $M_3$, $R_2$ and $R_3$ are
nodes. Considering the position  of the saddles $M_4$ and $R_1$ we
univocally get a phase portrait which is topologically equivalent
to {Picture 3.1(c3)}.

(b)   Assuming  $U_4>0$   we obtain
$\Delta_2\widetilde{\Delta}_2<0$ and
$\Delta_3\widetilde{\Delta}_3<0$. As $\widetilde{\Delta}_2>0$ we
obtain $\Delta _2<0$, i.e. the singular point $M_2$ is a saddle.
Therefore the other two finite singularities are nodes. Hence
$\Delta _3>0$ and this implies $\widetilde{\Delta}_3<0$, i.e. the
infinite point $R_3$ is a saddle.  i.e. the singular point $M_2$
is a node.  Considering   the location of the singularities we
obtain univocally   {Picture 3.1(c4)}.

 {\it II. The subcase  $B_3U_1 > 0$.}
 Considering Remark
\ref{rem:signs-U2,U4,B3U1,B3-nu} we obtain  $\Delta_1>0$,   i.e.
$M_1(0,0)$ is a node. Since by assumption  $R_2$ is a node (i.e.
$\widetilde{\Delta}_2>0$), by Remark
\ref{rem:signs-U2,U4,B3U1,B3-nu} we get $\operatorname{sign}(U_2)=
 \operatorname{sign}({\Delta}_2{\Delta}_3 \widetilde{\Delta}_3)$ and $\operatorname{sign}(B_3H_{14})=
\operatorname{sign}(\widetilde{\Delta}_3)$.


(1) {\it Assume first $U_2<0$.}  Then
${\Delta}_2{\Delta}_3 \widetilde{\Delta}_3<0$ and we shall
consider two subcases: $B_3H_{14}<0$ and $B_3H_{14}>0$.

(a)  If $B_3H_{14}<0$ then   we have $\widetilde{\Delta}_3<0$,
i.e. $R_3$ is a saddle and then $R_1$ is a node. In this case the
condition $U_2<0$ implies ${\Delta}_2{\Delta}_3>0$ and as $M_1$ is
a node, both  singular points  $M_2$ and $M_3$  must be saddles.
Therefore the fourth point $M_4$ is a node. So considering the
location  of the singular points and their respective  types we
univocally obtain    {Picture 3.1(c1)}.

(b) Suppose now   $B_3H_{14}>0$, i.e. $\widetilde{\Delta}_3>0$.
Hence both points $R_2$ and $R_3$ are nodes and $R_1$ is a saddle
and the condition $U_2<0$ yields ${\Delta}_2{\Delta}_3<0$, i.e.
one of the points $M_2$ or $M_3$ is a node and another one is a
saddle. Since $R_2$ and $R_3$ are nodes due to a substitution (see
Remark \ref{rem:change-sys-3.1}) we may consider that $M_2$ is a
saddle. So $M_4$   is a saddle and we arrive univocally at a phase
portrait topologically  equivalent to {Picture 3.1(c3)}.

(2) {\it Admit now $U_2>0$.}  Then ${\Delta}_2{\Delta}_3
\widetilde{\Delta}_3>0$  and according to Remark
\ref{rem:signs-U2,U4,B3U1,B3-nu}  we obtain $\operatorname{sign}(U_4)=
\operatorname{sign}({\Delta}_1 {\Delta}_2 \widetilde{\Delta}_2)$ and as
$\Delta_1>0$ and $\widetilde{\Delta}_2>0$ we get $\operatorname{sign}(U_4)=
 \operatorname{sign}({\Delta}_2)$.

(a)  If $U_4<0$ then we have ${\Delta}_2<0$ and
$\Delta_3\widetilde{\Delta}_3<0$.  So we   have two possibilities:
\emph{(i)} $\Delta_3<0$ (in which case   systems
\eqref{sys:Config.3.1-G} possess three     saddles ($M_2$, $M_3$,
$R_1$) and four nodes  ($M_1$, $M_4$,  $R_2$,  $R_3$)  and
\emph{(ii)} $\Delta_3>0$ (in this case  the three saddles are
 $M_2$, $M_4$ and $R_3$,  and the four  nodes are  $M_1$, $M_3$,
$R_1$ and $R_2$). Considering the location of these singularities
and their types, in both  cases we arrive at phase portraits which
are topologically equivalent to  {Picture 3.1(c3)}.

(b) Assuming  $U_4>0$   we obtain $ {\Delta}_2>0$ and
$\Delta_3\widetilde{\Delta}_3>0$. So $M_2$ is a node and then
$M_3$ and $M_4$ are saddles. This implies ${\Delta}_3<0$ and then
$\widetilde{\Delta}_3<0$, i.e. $R_3$ is a saddle and the remaining
infinite singular point $R_1$  must be a node.  In the same manner
as above, considering the types and the location of the
singularities we arrive at a phase portrait which is topologically
equivalent to {Picture 3.1(c1)}.

Summarizing the sets of conditions given above for each one of the
{Pictures 3.1(c\,j)}, $j=1,2,3,4$  we conclude that our
claim is proved.

\subparagraph{\it Suppose now  $W_4<0$ or  $W_4=0$ and $W_3<0$.}
So on the phase plane  systems \eqref{sys:Config.3.1-G} possess
two saddles, one node and one focus, and at infinity they have two
nodes and a saddle.  We assume again that $R_2$ is a node, i.e.
$\widetilde{\Delta}_2>0$.

We claim that the phase portrait of a system in this family   is
given by one of the ones indicated below if and only if the
corresponding conditions are satisfied, respectively:
\begin{gather*}
 \text{Picture 3.1($\stackrel{\,*}{c}$1)} \Leftrightarrow    B_3U_1 > 0,
U_2 < 0;  \\
 \text{Picture 3.1($\stackrel{\,*}{c}$2)} \Leftrightarrow   B_3U_1 < 0,
U_2 < 0;   \\
 \text{Picture 3.1($\stackrel{\,*}{c}$3)} \Leftrightarrow    B_3U_1 > 0,
U_2 > 0;  \\
 \text{Picture 3.1($\stackrel{\,*}{c}$4)}  \Leftrightarrow    B_3U_1 < 0,
U_2 > 0.
\end{gather*}
Indeed to convince ourselves we shall examine again   both cases:
$B_3U_1 < 0$ and $B_3U_1 > 0$.

 {\it I. The case  $B_3U_1 < 0$.}  Then  by Remark
\ref{rem:signs-U2,U4,B3U1,B3-nu} it follows $\Delta_1<0$ (yielding
$f<0$) (i.e. $M_1(0,0)$ is a saddle).

Since $\widetilde{\Delta}_2>0$ according to Remark
\ref{rem:signs-U2,U4,B3U1,B3-nu}     we obtain $\operatorname{sign}(U_2)=
 \operatorname{sign}( {\Delta}_2{\Delta}_3 \widetilde{\Delta}_3)$. Moreover,
since $M_4$ is a focus  then either  $M_2$ or  $M_3$ must be a
saddle, i.e. ${\Delta}_2{\Delta}_3<0$ and this implies
  $\operatorname{sign}(U_2)= -\operatorname{sign}(\widetilde{\Delta}_3)$.

(1) {\it Assume first $U_2<0$.}  Then $R_3$ is a node and
$R_1$ is a saddle. Taking into account that $R_2$ and $R_3$ are
both nodes then without loss of generality, due to Remark
\ref{rem:change-sys-3.1} we may assume that  $M_2$ is a saddle
(then $M_3$ is a node).  So considering the types and the location
of all the singularities we arrive at {Picture
3.1($\stackrel{\,*}{c}$2)}.

(2) {\it Admit now $U_2>0$.} In this case
$\widetilde{\Delta}_3<0$ ($h<0$), i.e. $R_3$ is a saddle and hence
$R_1$ is a node. We observe that the conditions $f<0$ and $h<0$
imply $fh-f-h>0$ and therefore $\Delta_3=f(fh-f-h)/h>0$, i.e.
$M_3$ is a node. Therefore $M_2$ is a saddle and as $M_4$ is a
focus we arrive at {Picture 3.1($\stackrel{\,*}{c}$4)}.

 {\it  II. The case  $B_3U_1>0$.} Herein by Remark
\ref{rem:signs-U2,U4,B3U1,B3-nu} it follows that $\Delta_1>0$
($f>0$),  i.e. $M_1(0,0)$ is a node. Since   $M_4$ is a focus then
clearly    $M_2$ and  $M_3$ should be saddles, i.e.  $\Delta_2<0$
and $\Delta_3<0$. Hence taking into account our assumption (i.e.
that $R_2$ is a node) according to Remark
\ref{rem:signs-U2,U4,B3U1,B3-nu} in this case we have $\operatorname{sign}(U_2)=
\operatorname{sign}(\widetilde{\Delta}_3)$.

If  $U_2<0$ we get $\widetilde{\Delta}_3<0$ (i.e. $R_3$ is a
saddle) and then $R_1$ is a node. This leads to {Picture
3.1(\,$\stackrel{\,*}{c}$1)}.

The condition   $U_2>0$ implies that both points    $R_2$ and
$R_3$ are nodes (then $R_1$ is a saddle) and we get
{Picture 3.1(\,$\stackrel{\,*}{c}$3)}. This completes the
proof of our claim and thus all the phase portraits associated with
{Config. 3.1}  as well as the respective invariant criteria
for the realization of each of them are determined.

\subsubsection{The phase portraits associated with
{Config. 3.2}} According to Table 2 we  consider the family of systems
\begin{equation}\label{sys:Config.3.2-G}
\dot x = x [1   + gx  + (h-1)y ], \quad \dot y  = y [  (g-1)x  + hy ],
\end{equation}
for which the condition
\begin{equation}\label{cond:A2}
gh(g+h-1)(g-1)(h-1)\ne0
\end{equation}
holds. We observe that this family of systems is a particular case
of the family  \eqref{sys:Config.3.1-G} when the parameter $f$
equals zero (and in this case the point $M_3$ has coalesced   with
$M_1(0,0))$.  For the    finite singularities of systems
\eqref{sys:Config.3.2-G} with the condition \eqref{cond:A2} we
have
\begin{equation}\label{expr:Delta-i,delta-i:3.2}
\begin{gathered}
M_3\equiv M_{1}(0,0):\quad \Delta_{1,3}=0, \quad \rho_{1,3}=1,\;  \delta_{1,3}=1;\\
M_2(-1/g,0):\quad \Delta_2=(g-1)/g, \quad \rho_{2}=(1-2g)/g,\; \delta_2=1/g^2;\\
M_4\left(\frac{h}{1-g-h},   \frac{1-g}{1-g-h}\right):\quad
\Delta_4=\frac{ -h (g-1)}{g+h-1}, \;
   \rho_4=\frac{h}{1-g-h},\; \delta_4=\rho_4^2-4\Delta_4.
\end{gathered}
\end{equation}
and for the three infinite singular points we have again
\begin{equation}\label{ISPs:tilde delta-i:3.2}
 R_1(1, 1, 0):\  \widetilde{\Delta}_1= 1 -g- h ;\quad
  R_2(1, 0, 0): \  \widetilde{\Delta}_2 = g;\quad
   R_3(0, 1, 0): \ \widetilde{\Delta}_3 = h.
\end{equation}
We observe that due to the relation $\rho_1=1$ (i.e. only one of
the respective eigenvalues vanishes) the double singular point
$M_{1}(0,0)$ is a saddle-node.


Taking into account  \eqref{expr:Delta-i,delta-i:3.2} and
\eqref{ISPs:tilde delta-i:3.2} we evaluate for systems
\eqref{sys:Config.3.2-G} the   invariant polynomials we need:
\begin{equation}\label{val:inv-Conf-3.2}
\begin{gathered}\mu_0= g h (g + h-1)=-\widetilde{\Delta}_1\widetilde{\Delta}_2\widetilde{\Delta}_3,\\
     K= 2 g(g-1) x^2 + 4 g h x y + 2 h(h-1)y^2,\\
W_4 = h^3 [4(g-1)^2 - 3 h + 4 g h]=\mu_0^2\,\delta_1\delta_2\delta_3\delta_4,\\
H_1= 288 h=288\widetilde{\Delta}_3,\\
H_5= 384(1-g)h^2 =-384{\Delta}_2 \widetilde{\Delta}_2 \widetilde{\Delta}_3^2,\\
H_{14}= 30 g h^2 =30 \widetilde{\Delta}_2 \widetilde{\Delta}_3^2,\\
 B_3= 3 h x^2 y^2=3\widetilde{\Delta}_1\rho_4 x^2y^2=3\widetilde{\Delta}_3 x^2y^2.
\end{gathered}
\end{equation}

\paragraph{The case $\mu_0<0$.}
In the same manner as in the previous section (in the case of
{Config. 3.1})  we   shall consider two subcases: $K<0$ and
$K>0$.

\subparagraph{The subcase $K<0$.} In this case for systems
 \eqref{sys:Config.3.2-G},  according to \cite{ArtLliVul08-IJBCh}
(see Table 1) on the finite part of the phase plane besides the
saddle-node there are two saddles.  We claim that  in this case we
obtain the unique phase portrait given by {Picture
3.2(d1)}.

Indeed,  due to the index theory    all three simple infinite
singularities \eqref{ISPs:tilde delta-i:3.2} are nodes.
Considering the position of the saddles  $M_2$ and $M_4$ and of
the saddle-node $M_{1}$  we get univocally   the phase portrait
given by {Picture 3.2(d1)}.

\subparagraph{The subcase $K>0$.} Following
\cite{ArtLliVul08-IJBCh} (see Table 1) we find  that besides the
saddle-node there are two anti-saddles. Moreover,   by
\eqref{expr:Delta-i,delta-i:3.2} and \eqref{val:inv-Conf-3.2} we
observe that the relation $W_4=0$ holds if and only if
$\delta_4=0$. So according to \cite{ArtLliVul08-IJBCh} (see Table
1) we have two nodes if $W_4\ge0$ and we have one node and one
focus if $W_4<0$.

On the other hand  in both cases we have two saddles and a node at
infinity.

 {\it I. Assume first $W_4\ge0$.} According to
\eqref{val:inv-Conf-3.2} we have
$\operatorname{sign}(H_1)=\operatorname{sign}(\widetilde{\Delta}_3$).


(1) If $H_1<0$  then $\widetilde{\Delta}_3<0$ (i.e. $h<0$)
and hence the singular point $R_3$ is a saddle. We claim that the
condition $W_4\ge0$ implies $\widetilde{\Delta}_2>0$, i.e. $g>0$.
Indeed  assuming the contrary  we have $g<0$ and due to $h<0$ we
obtain  $h[4(g-1)^2 - 3 h + 4 g h]<0$ which contradicts $W_4\ge0$
(see \eqref{val:inv-Conf-3.2}).

Thus  $g>0$ (i.e. $\widetilde{\Delta}_2>0$) and then $R_2$ is a
node and consequently $R_3$ must be a saddle. Considering the
position and the types  of the singularities we arrive at
{Picture 3.2(e2)}.

(2) Suppose now $H_1>0$.  Then $\widetilde{\Delta}_3>0$
(i.e. $h>0$) and hence the singular point $R_3$ is a node and
consequently the other two infinite points are saddles.  Obviously
we get  {Picture 3.2(e1)}.

 {\it II.  Admit now  $W_4<0$.} Then  the point $M_4$
is a focus and $M_2$ is a node, i.e. $\Delta _2>0$.

(1) If $H_1<0$  then $\widetilde{\Delta}_3<0$ (i.e. $h<0$)
and hence the singular point $R_3$ is a saddle. According to
\eqref{val:inv-Conf-3.2} due to $\Delta _2>0$ we obtain
 $\operatorname{sign}(H_5)=-\operatorname{sign}(\widetilde{\Delta}_2$).

(a) Assume $H_5<0$. Then $\widetilde{\Delta}_2>0$ and hence $R_2$
is a node. So the remaining infinite point $R_1$ must be a saddle.
As $M_{1}(0,0)$ is a saddle-node and $M_2$ is a node we get
{Picture 3.2($\stackrel{\,*}{e}$2)}.

(b) If $H_5>0$ then $\widetilde{\Delta}_2<0$ and the singular
point $R_2$ is a saddle, whereas $R_1$ is a node. As $g<0$, $h<0$
and $1-g-h>0$ considering \eqref{expr:Delta-i,delta-i:3.2} we
obtain
$$
\rho_2\rho_4=\frac{(1-2g) h}{g (1 - g - h)}>0.
$$
 Hence   we conclude  that the node $M_2$ and the focus $M_4$
have the same stability  and we univocally obtain {Picture
3.2($\stackrel{\,*}{e}$3)}.

(2) Assume now  $H_1>0$.  Then we have
$\widetilde{\Delta}_3>0$ (i.e. $h>0$) and the singular point $R_3$
is a node. Therefore the remaining two infinite singular points
are saddles and considering the location of the singularities we
obtain univocally  {Picture 3.2($\stackrel{\,*}{e}$1)}.

\paragraph{The case $\mu_0>0$.} According to
\cite{ArtLliVul08-IJBCh} (see Table 1) the systems
\eqref{sys:Config.3.2-G},besides the saddle-node  possess one
saddle and in addition either one node if $W_4\ge0$, or one focus
if $W_4<0$. On the other hand at infinity we have a saddle and two
nodes.

\subparagraph{The subcase $W_4\ge0$.} We claim that in this case
the phase portrait of  a  system   \eqref{sys:Config.3.2-G} is
necessarily one of  those indicated below if and only if the
conditions on the right side are satisfied:
\begin{gather*}
\text{Picture 3.2(f1)}  \Leftrightarrow   B_3H_{14}<0, H_5<0; \\
\text{Picture 3.2(f2)}  \Leftrightarrow   B_3H_{14}<0, H_5>0, H_1<0; \\
\text{Picture 3.2(f3)}  \Leftrightarrow   B_3H_{14}<0, H_5>0, H_1>0; \\
\text{Picture 3.2(f4)}   \Leftrightarrow   B_3H_{14}>0, H_5<0; \\
\text{Picture 3.2(f5)}  \Leftrightarrow   B_3H_{14}>0,   H_5>0.
\end{gather*}


Indeed, first of all we observe that according to
\eqref{val:inv-Conf-3.2} we have:
\begin{equation}\label{sign:B3-nu,H1}
\operatorname{sign}( B_3H_{14})=\operatorname{sign}(\widetilde{\Delta}_2\widetilde{\Delta}_3),
\quad \operatorname{sign}(H_1)=\operatorname{sign}(\widetilde{\Delta}_3),\quad
\operatorname{sign}(H_5)=-\operatorname{sign}({\Delta}_2\widetilde{\Delta}_2),
\end{equation}
so we can control the signs of each of the   determinants
$\widetilde{\Delta}_2$, $\widetilde{\Delta}_3$ and ${\Delta}_2$.

 {\it I. The possibility $ B_3H_{14}<0$.}  Then
$\widetilde{\Delta}_2\widetilde{\Delta}_3<0$ and we shall consider
two subcases: $H_5<0$ and $H_5>0$.

(1) If $H_5<0$  then considering \eqref{sign:B3-nu,H1} and
\eqref{expr:Delta-i,delta-i:3.2} we obtain $g^2(g-1)>0$, i.e.
$g>1$. Consequently $\widetilde{\Delta}_2>0$ and
$\widetilde{\Delta}_3<0$. Hence the singular points $R_2$, $R_1$
and $M_2$ are   nodes, whereas  $R_3$ and $M_4$ are saddles.
Considering the position  of these singularities we univocally get
{Picture 3.2(f1)}.

(2) Assume $H_5>0$. Then
${\Delta}_2\widetilde{\Delta}_2<0$ and we consider two
possibilities: $H_1<0$ and $H_1>0$.

(a) Assume first $H_1<0$. According to \eqref{sign:B3-nu,H1} we
have $\widetilde{\Delta}_3<0$ and hence we obtain
$\widetilde{\Delta}_2>0$  which implies ${\Delta}_2<0$. So besides
the saddle-node $M_{1}(0,0)$ systems \eqref{sys:Config.3.2-G} have
the nodes $R_2$, $R_1$ and $M_4$ and the saddles $R_3$ and $M_2$.
This obviously leads to the phase portrait given by
{Picture 3.2(f2)}.


(b) If $H_1>0$, similarly as above, we get the nodes $R_3$, $R_1$
and $M_2$ and the saddles $R_2$ and $M_4$. Considering the
location  of these singularities we univocally obtain
{Picture 3.2(f3)}.

 {\it II.  The possibility $ B_3H_{14}>0$.}  In this
case we have $\widetilde{\Delta}_2\widetilde{\Delta}_3>0$ and as
there could not be two saddles at infinity,   both points $R_2$
and $R_3$ are nodes (i.e. $\widetilde{\Delta}_2>0$ and
$\widetilde{\Delta}_3>0$), whereas the point $R_1$ is a saddle.

(1) If $H_5<0$ then by \eqref{sign:B3-nu,H1}   we obtain
${\Delta}_2>0$ (then $g>1$)  and therefore $M_2$ is a node and
$M_4$ is a saddle. Considering the location of all the
singularities we univocally obtain {Picture 3.2(f4)}.

(2) Assume $H_5>0$.  This implies   $ {\Delta}_2<0$  (then
$0<g<1$)  and hence $M_2$ is a saddle and $M_4$ is a node. Thus we
get {Picture 3.2(f5)}.

\subparagraph{The subcase $W_4<0$.} As we mentioned above, in this
case systems \eqref{sys:Config.3.2-G} have a saddle and a focus
besides the saddle-node. Clearly a focus   could only be the
singularity $M_4$ and hence  $M_2$ is a saddle. Therefore we have
$\Delta_2<0$ which implies $ 0<g<1$ by
\eqref{expr:Delta-i,delta-i:3.2}. Consequently
$\widetilde{\Delta}_2>0$, i.e. $R_2$ is a node. It remains to
distinguish the possibilities when $R_3$ is a node or a saddle.
According to \eqref{sign:B3-nu,H1}, these situations are governed
by the invariant polynomial $H_1$. More precisely, we obtain the
saddle $R_3$ and the node $R_1$ if $H_1 < 0$, and  we have the
saddle $R_1$ and the node $R_3$ if $H_1 > 0$. Considering the
location of the singularities in the first case we obtain
{Picture 3.2($\stackrel{\,*}{f}$2)}, whereas in the second
case we have {Picture 3.2($\stackrel{\,*}{f}$5)}.


\subsubsection{The phase portraits associated with {Config.
3.3}} According to  Table 2 we shall consider the family of
systems
\begin{equation}\label{sys:Config.3.3-G}
\dot x = x [g   + gx  + (h-1)y ], \quad \dot y  = y [g-1  +(g-1)x  + hy ],
\end{equation} for which the condition
\begin{equation}\label{cond:A2-a}
gh(g+h-1)(g-1)(h-1)\ne0
\end{equation}
holds and we  keep the same notations for the singularities. For
systems  \eqref{sys:Config.3.3-G} we have:
\begin{equation}\label{expr:Delta-i,delta-i:3.3}
\begin{gathered} M_1(0,0):  \Delta_1=g(g-1);\ \delta_1=g^2;\\
M_4\equiv M_2(-1,0): \  \Delta_2=0, \rho_2=-g, \     \delta_2=g^2;\\
M_3 \big(0, (1-g)/h\big): \ \Delta_3= (g-1)(1-g-h)/h,  \ \
\delta_3=  (gh-g-1)^2/h^2
\end{gathered}\end{equation} for the finite singularities and
\begin{equation}\label{ISPs:tilde delta-i:3.3}
 R_1(1, 1, 0):\  \widetilde{\Delta}_1= 1 -g- h ;\quad
  R_2(1, 0, 0): \  \widetilde{\Delta}_2 = g;\quad
   R_3(0, 1, 0): \ \widetilde{\Delta}_3 = h.
\end{equation} for the infinite ones. We observe that due to the condition
\eqref{cond:A2-a} we have $\rho_2=-g\ne0$ (i.e. only one of the
corresponding eigenvalues vanishes) and hence the double singular
point $M_{1}(0,0)$ is a saddle-node.


Considering the expressions above for systems
\eqref{sys:Config.3.3-G} we obtain
\begin{equation}\label{val:Conf-3.3}
\begin{gathered}
\mu_0= g h (g + h-1)=-\widetilde{\Delta}_1\widetilde{\Delta}_2\widetilde{\Delta}_3,\\
     K= 2 g(g-1) x^2 + 4 g h x y + 2 h(h-1)y^2,\\
 B_3= 3g(g+h-1)x^2y^2= -3\widetilde{\Delta}_1\widetilde{\Delta}_2x^2y^2,\\
H_1= 288g(g+h-1)=-288\widetilde{\Delta}_1 \widetilde{\Delta}_2,\\
H_5= 384 (1 - g) g^2 (1 -g - h)^2
=-384 \Delta_1\widetilde{\Delta}_2\widetilde{\Delta}_1^2.\\
 H_{14}= g^2 h (g+h-1)
 =-30  \widetilde{\Delta}_1 \widetilde{\Delta}_2^2 \widetilde{\Delta}_3,
\end{gathered}
\end{equation}
Clearly   the next relations hold:
\begin{equation}\label{sign:B3-nu,H1-1}
\begin{gathered}
\operatorname{sign}( B_3H_{14})
 =\operatorname{sign}(\widetilde{\Delta}_2\widetilde{\Delta}_3),
\quad \operatorname{sign}(H_1)=-\operatorname{sign}(\widetilde{\Delta}_1
\widetilde{\Delta}_2),\\
\operatorname{sign}(H_5)=-\operatorname{sign}({\Delta}_1\widetilde{\Delta}_2).
\end{gathered}
\end{equation}
Since $\mu_0=-\widetilde{\Delta}_1 \widetilde{\Delta}_2)
\widetilde{\Delta}_3)$ we can control the signs of each one of the
determinants $\widetilde{\Delta}_2$,  $\widetilde{\Delta}_3$ and
${\Delta}_1$.

 Since in this case we have one double (a saddle-node) and two simple
finite singularities, the types of these points are determined by
the same conditions   indicated in the previous section (for
systems \eqref{sys:Config.3.2-G}). So we shall consider the same
cases as for {Config. 3.2}.

\paragraph{The case $\mu_0<0$.}  According to
\cite{ArtLliVul08-IJBCh} (see Table 1) the simple   singular
points $M_1$ and $M_3$ are of the same type.

\subparagraph{The subcase $K<0$.} Then $M_1$ and $M_3$ are saddles
and all the infinite points are nodes and this univocally leads to
the phase portrait given by {Picture 3.3(d1)}.

\subparagraph{The subcase $K>0$.} Then $M_1$ and $M_3$ are nodes
and at infinity we have two saddles and one node. In order to
distinguish which one among the infinite points is a node, we
apply the invariant polynomial   $H_5$ considering its sign given
in \eqref{sign:B3-nu,H1-1}.

 {\it I. Assume first $H_5<0$.} Then due to $
{\Delta}_1>0$ (as  $M_1$ is a node) we obtain
$\widetilde{\Delta}_2>0$  and hence  $R_2$ is a node. Consequently
$R_3$ and $R_1$ are saddles and we univocally obtain
{Picture 3.3(e1)}.


 {\it II. Suppose now  $H_5>0$.}  In this case
$\widetilde{\Delta}_2<0$ (i.e. $g<0$) and the singular point $R_2$
is a saddle. So we have two possibilities:  $h<0$ (when $R_3$ is a
saddle and $R_1$ is a node) and  $h>0$ (when $R_3$ is a node and
$R_1$ is a saddle). It easily could be determined that in both
cases we obtain phase portraits  topologically equivalent to
{Picture 3.3(e2)}.

\paragraph{The case $\mu_0>0$.}  Since both   singular points
$M_1$ and $M_3$ are located on the invariant lines we conclude
that one of them is a saddle and another one is a node. Regarding
the infinite singular points, clearly by the index theory there
should be a saddle and two nodes.  We claim that in this case the
phase portrait of a system \eqref{sys:Config.3.3-G} is one of
those  indicated below if and only if the corresponding conditions
on the right side are satisfied:
\begin{gather*}
\text{Picture 3.3(f1)}  \Leftrightarrow   B_3H_{14}<0, H_5<0; \\
\text{Picture 3.3(f2)}  \Leftrightarrow   B_3H_{14}<0, H_5>0, H_1<0; \\
\text{Picture 3.3(f3)}  \Leftrightarrow   B_3H_{14}<0, H_5>0, H_1>0; \\
\text{Picture 3.3(f4)}   \Leftrightarrow   B_3H_{14}>0, H_5<0; \\
\text{Picture 3.3(f5)}  \Leftrightarrow   B_3H_{14}>0,   H_5>0.
\end{gather*}
Indeed in order to prove this claim, considering
\eqref{sign:B3-nu,H1-1} we shall examine two cases.

\subparagraph{The subcase $B_3H_{14}<0$.} Then
$\widetilde{\Delta}_2\widetilde{\Delta}_3<0$ and as $\mu_0>0$
according to \eqref{val:Conf-3.3} it follows
$\widetilde{\Delta}_1>0$, i.e. $R_1$ is a node.


 {\it I. Assume first $H_5<0$.} In this case by
\eqref{sign:B3-nu,H1-1} we obtain $\Delta_1\widetilde{\Delta}_2>0$
and this implies $g^2(g-1)>0$. Therefore $g>1$    and consequently
$\widetilde{\Delta}_1>0$ and $\widetilde{\Delta}_3<0$. Hence the
singular points $R_2$, $R_1$ and $M_1$ are   nodes, whereas $R_3$
and $M_3$ are saddles. Considering the location  of these
singularities we univocally get {Picture 3.3(f1)}.

 {\it II. Suppose now $H_5>0$.} Then $g-1<0$ and we
consider two possibilities:   $H_1<0$ and $H_1>0$.

(1) If $H_1<0$, then  considering \eqref{sign:B3-nu,H1-1}
we obtain $\operatorname{sign}(B_3H_{14}H_1)= \operatorname{sign}(\mu_0
\widetilde{\Delta}_2)<0$ and as $\mu_0>0$ (i.e.
$\widetilde{\Delta}_1\widetilde{\Delta}_2\widetilde{\Delta}_3<0$)
we get   $\widetilde{\Delta}_2>0$. Therefore the condition $H_1<0$
yields  $\widetilde{\Delta}_1>0$ and the condition $H_5>0$ implies
${\Delta}_1<0$. Thus   besides the saddle-node $M_2(0,0)$, systems
\eqref{sys:Config.3.3-G} have the nodes $R_2$, $R_1$ and $M_3$ and
the saddles $R_3$ and $M_1$. This obviously leads to the phase
portrait given by {Picture 3.3(f2)}.

(2) If $H_1>0$ then in a similar way as above we get the
nodes $R_3$, $R_1$ and $M_1$ and the saddles $R_2$ and $M_3$.
Considering the locations of these singularities we univocally
obtain {Picture 3.3(f3)}.

\subparagraph{The subcase $B_3H_{14}>0$.}  Then
$\widetilde{\Delta}_2\widetilde{\Delta}_3>0$ and as at infinity
there could not be two saddles  we obtain that both points $R_2$
and $R_3$ are nodes (i.e. $\widetilde{\Delta}_2>0$ and
$\widetilde{\Delta}_3>0$), whereas the point $R_1$ is a saddle.
Therefore considering \eqref{sign:B3-nu,H1-1} we have
$\operatorname{sign}(H_5)=-\operatorname{sign}({\Delta}_1)$, i.e. $H_5$ governs the types of
the finite singularities. It is clear that we get {Picture
3.3(f4)} if $H_5<0$ and {Picture 3.3(f5)} if $H_5>0$.


\subsubsection{The phase portraits associated with {Config.
3.4}} Considering   Table 2 we   examine the family of systems
\begin{equation}\label{sys:Config.3.4-G}
\dot x = x [1  + (h-1)y ], \quad \dot y  = y(f - x+hy),
\end{equation} for which the condition
\begin{equation}\label{cond:A3}
h(h-1)f(f-1)(f+h-fh)\ne0
\end{equation}
holds. We observe that this family of systems is a particular case
of the family  \eqref{sys:Config.3.1-G} when the parameter $g$
equals zero  and in this case the point $M_2$ has gone to infinity
and  has coalesced  with $R_2(1, 0, 0)$.  So these systems possess
three simple finite singularities and three infinite singularities
(one of which being double).   For all the singularities  of these
systems with the condition \eqref{cond:A3} we have:
\begin{equation}\label{expr:Delta-i,delta-i:3.4}
\begin{gathered}
M_1(0,0):\; \Delta_1=f,  \;  \delta_1=(1-f)^2;\\
M_3(0,-f/h):\;   \Delta_3=f(fh-f-h)/h, \; \delta_3= {(f + h)^2}/{h^2};\\
M_4\left( \frac{f+h-fh}{1-h},  \frac{1}{1-h}\right):\;
\Delta_4= \frac{fh-f-h}{1-h}, \;  \rho_4=\frac{h}{1-h}, \;
  \delta_4=\rho_4^2-4\Delta_4
\end{gathered}
\end{equation}
for the finite singularities and
\begin{equation}\label{ISPs:tilde delta-i:3.4}
 R_1(1, 1, 0):\  \widetilde{\Delta}_1= 1-h;\ \
  R_2(1, 0, 0)  :\  \widetilde{\Delta}_2 = 0, \rho_2=1;\ \
   R_3(0, 1, 0) :\  \widetilde{\Delta}_3 = h
\end{equation}
for the infinite ones. As $\rho_2\ne0$ the singular point $R_2(1,
0, 0)$ is a saddle-node (both hyperbolic sectors being on the same
part of the infinite line).


Considering \eqref{expr:Delta-i,delta-i:3.4} and \eqref{ISPs:tilde
delta-i:3.4} for systems \eqref{sys:Config.3.4-G} we evaluate  the
 invariant polynomials which we need:
\begin{equation}\label{val:conf-3.4}
\begin{gathered}
\mu_0=0,\quad K=  2h(h-1)y^2 ,\quad  \eta= 1,\\
 B_3= 3h(1-f)x^2y^2=3(1-f)\widetilde{\Delta}_3x^2y^2,\\
  W_3= \widetilde{\Delta}_1^2 \widetilde{\Delta}_3^2(\delta_1\,\delta_3+
\delta_1\,\delta_4+\delta_3\delta_4),\\
W_4 = \widetilde{\Delta}_1^2 \widetilde{\Delta}_3^2\,\delta_1
\delta_3\delta_4,\\
 U_{1}= \frac{1}{8}fh (1 - f) (1 - h)^2 =
   \frac{1}{8}\Delta_1 \widetilde{\Delta}_1^2\widetilde{\Delta}_3(1-f),\\
 U_{2}=  3f(f+h-fh) = -3\Delta_3\widetilde{\Delta}_3,\\
H_5= -384\big[h(fh-f-h)+f^2(h-1)\big],\\
G_9= h(h-1)/8=-\widetilde{\Delta}_1\widetilde{\Delta}_3/8.
\end{gathered}
\end{equation}
From this, considering \eqref{expr:Delta-i,delta-i:3.4} and
\eqref{ISPs:tilde delta-i:3.4} we get the following relations:
\begin{equation}\label{signs:conf-3.4}
   \operatorname{sign}(B_3U_1) = \operatorname{sign}(\Delta_1);\quad
  \operatorname{sign}(U_2)=  -\operatorname{sign}( \Delta_3 \widetilde{\Delta}_3).
\end{equation}
For systems  \eqref{sys:Config.3.4-G} with three finite simple
singularities we have  $\mu_0=0$ and  $G_9=h(h-1)/8$.
 Therefore considering the fact that we could not have
two foci, according to \cite{ArtLliVul08-IJBCh} (see Table 1) the
types of the finite singularities are determined by the
corresponding affine invariant conditions on the right side:
\begin{equation}\label{types:sing-3.4}
\begin{gathered}
(g)\ \tilde s,\tilde s,\tilde n\quad \Leftrightarrow\quad   K<0,
W_4\ge0;\\
(\stackrel{*}{g})\ \tilde s,\tilde s,\tilde f\quad  \Leftrightarrow \quad K<0, W_4<0;\\
(h)\ \tilde s,\tilde n,\tilde n\quad\Leftrightarrow \quad K>0
\text{ and either }
 W_4>0, \text{ or }  W_4=0, W_3\ge0;\\
(\stackrel{*}{h})\ \tilde s,\tilde n,\tilde f\quad\Leftrightarrow
\quad K>0 \text{ and either } W_4<0, \text{ or }\  W_4=0, W_3<0;
\end{gathered}
\end{equation}

So we  consider the two cases: $K<0$ and $K>0$.

\paragraph{The case $K<0$.} Then by \eqref{val:conf-3.4} we have $0<h<1$ and
considering \eqref{ISPs:tilde delta-i:3.4} we get
$\widetilde{\Delta}_1>0$ and $\widetilde{\Delta}_3>0$. So apart
from the saddle-node $R_2(1, 0, 0)$,  systems
\eqref{sys:Config.3.4-G} possess at the infinity   two nodes.

\subparagraph{The subcase $W_4\ge0$.} According to
\eqref{types:sing-3.4} we have two saddles and one node. We claim,
that in this case the phase portrait of a system
\eqref{sys:Config.3.4-G} corresponds to one of those indicated
below if and only if the corresponding conditions on the right
side are satisfied:
\begin{gather*}
\text{Picture 3.4(g1)}  \Leftrightarrow   B_3U_1<0, U_2<0; \\
\text{Picture 3.4(g2)}  \Leftrightarrow    B_3U_1<0, U_2>0; \\
\text{Picture 3.4(g3)}  \Leftrightarrow   B_3U_1>0.
\end{gather*}
Indeed, considering \eqref{signs:conf-3.4} we  examine two
possibilities: $B_3U_1<0$ and $B_3U_1>0$.

 {\it I. The possibility  $B_3U_1<0$.} Then
$\Delta_1<0$ and the singular point $M_1(0,0)$ is a saddle.

 If $U_2<0$ then considering \eqref{signs:conf-3.4} and
the relation $\widetilde{\Delta}_3>0$ we obtain ${\Delta}_3>0$.
Hence $M_3$ is a node and consequently   $M_4$ is a saddle and
this leads univocally to {Picture 3.4(g1)}.

Assume now $U_2>0$.  In this case we get ${\Delta}_3<0$ and then
$M_3$ is a saddle whereas $M_4$ is a node. In this case we
univocally get the phase portrait given by
{Picture 3.4(g2)}.

 {\it II. The possibility  $B_3U_1>0$.} In this case
we obtain that   the singular point $M_1(0,0)$ is a node (as
$\Delta_1>0$). Hence the other two singularities are saddles and
considering also the infinite singularities  we obtain
{Picture 3.4(g3)}.

\subparagraph{The subcase $W_4<0$.} As a focus could  only  be at
the singular point $M_4$ we obtain that $M_1$ and $M_3$ are
saddles and considering the nodes $R_1$ and $R_3$ at infinity this
univocally leads to {Picture 3.4($\stackrel{\,*}{g}$2)}.

\paragraph{The case $K>0$.} Then $h(h-1)>0$ and considering
\eqref{ISPs:tilde delta-i:3.4} we get
$\widetilde{\Delta}_1\widetilde{\Delta}_3<0$, i.e. at infinity
besides the saddle-node we have one saddle and one node.

On the other hand  for systems \eqref{sys:Config.3.4-G} according
to \eqref{types:sing-3.4}, on the phase plane there exist one
saddle and two anti-saddles and these two possibilities are
distinguished by the invariant polynomials $W_4$ and $W_3$ as it
is indicate in \eqref{types:sing-3.4}.

\subparagraph{The subcase $W_4>0$ or $W_4=0$ and $W_3\ge0$.} Then
we have one saddle and two nodes. We claim that in this case the
phase portrait of a system \eqref{sys:Config.3.4-G} is given by
one of those indicated below if and only if the corresponding
conditions are satisfied, respectively:
\begin{gather*}
\text{Picture 3.4(h1)}\ \Leftrightarrow\  B_3U_1<0; \\
\text{Picture 3.4(h3)}\ \Leftrightarrow\  B_3U_1>0,  U_2<0, H_5>0; \\
\text{Picture 3.4(h2)}\ \Leftrightarrow\ B_3U_1>0 \text{ and either }
  U_2>0, \text{ or } U_2<0, H_5<0.
\end{gather*}
To prove this claim we shall consider again two possibilities:
$B_3U_1<0$ and $B_3U_1>0$.

 {\it I. The possibility  $B_3U_1<0$.}  Then $
{\Delta}_1 <0$ (i.e. $f<0$) and $M_1$ is a saddle whereas the
other  two points are nodes. We observe that due to $\delta_4\ge0$
the condition $K>0$ (i.e. $h(h-1)>0$) implies $h>1$. Indeed since
$f<0$, supposing $h<0$    we clearly obtain a contradiction:
$\delta_4=\frac{4f(h-1)^2-3h^2+4 h}{(h-1)^2}<0$. Thus, $h>1$ and
then $R_3$ is a node and $R_1$ is a saddle. This immediately leads
to the {Picture 3.4(h1)}.

 {\it II. The possibility  $B_3U_1>0$.} In this case
we obtain $ {\Delta}_1 >0$ (i.e. $f>0$) and $M_1$ is a node. Since
by \eqref{signs:conf-3.4}  the invariant polynomial $U_2$ governs
the sign of the product $\Delta_3 \widetilde{\Delta}_3$, we
examine two subcases: $U_2<0$ and $U_2>0$.

(1) Assume first $U_2>0$. Then by \eqref{signs:conf-3.4}
we obtain ${\Delta}_3\widetilde{\Delta}_3<0$, i.e. the singular
points $M_3$ and $R_3$ are of different types. Fixing first
${\Delta}_3>0$ and, secondly  ${\Delta}_3<0$, the types of all the
singularities,  as well as their location  become well determined.
In both cases we get phase portraits which are topologically
equivalent to {Picture 3.4(h2)}.

(2) Admit now $U_2<0$.  Then we obtain
${\Delta}_3\widetilde{\Delta}_3>0$ and we have to distinguish via
invariant polynomials when these determinants are both negative
and when they are positive. Due to the condition $f>0$ and
considering \eqref{val:conf-3.4}, \eqref{expr:Delta-i,delta-i:3.4}
and the relation $\widetilde{\Delta}_1\widetilde{\Delta}_3<0$  we
obtain
\begin{align*}
&\operatorname{sign}({\Delta}_3)= \operatorname{sign}\big(h(fh-f-h)\big)=
\operatorname{sign}(\widetilde{\Delta}_3)= -\operatorname{sign}(\widetilde{\Delta}_1)=\operatorname{sign}\big(f^2(h-1)\big)\\
&\Rightarrow\
\operatorname{sign}({\Delta}_3)=\operatorname{sign}\big(h(fh-f-h)+f^2(h-1)\big)= -\operatorname{sign}(H_5).
\end{align*}

So if $H_5<0$ then ${\Delta}_3>0$, $\widetilde{\Delta}_3>0$ and
$\widetilde{\Delta}_1<0$. Hence systems \eqref{sys:Config.3.4-G}
possess  the nodes $M_1$, $M_3$ and $R_3$ and the saddles $M_4$
and $R_1$ ($R_2$ being a saddle-node). Therefore we get a phase
portrait which is topologically equivalent to {Picture
3.4(h2)}.

In the case  $H_5>0$ in the same manner as above we get the nodes
$M_1$, $M_4$ and $R_1$ and the saddles $M_3$ and $R_3$. This leads
univocally to  {Picture 3.4(h3)}.

\subparagraph{The subcase $W_4<0$ or $W_4=0$ and $W_3<0$.} Then
systems \eqref{sys:Config.3.4-G}  have one saddle, one node and
one focus.   We claim that   the phase portrait of a system in
this family  corresponds to one of those indicated below if and
only if the  corresponding conditions are satisfied:
\begin{equation}\label{cond:portraits-3.4}
\begin{gathered}
\text{Picture 3.4($\stackrel{\,*}{h}$1)}\ \Leftrightarrow\
        B_3U_1<0, U_2<0;  \\
\text{Picture 3.4($\stackrel{\,*}{h}$2)}\ \Leftrightarrow\
        B_3U_1>0, U_2>0; \\
\text{Picture 3.4($\stackrel{\,*}{h}$3)}\ \Leftrightarrow\
        B_3U_1>0, U_2<0; \\
\text{Picture 3.4($\stackrel{\,*}{h}$4)}\ \Leftrightarrow\  B_3U_1<0, U_2>0.
\end{gathered}
\end{equation}
Indeed, first we mention that   the focus could   only  be the
singularity $M_4$ and that the type of the singularity $M_1$ is
governed again by the invariant polynomial $B_3U_1$. Moreover, we
observe that in the case under examination we have
$\widetilde{\Delta}_1\widetilde{\Delta}_3<0$ and
$\Delta_1\Delta_3<0$ (since one of the  singularities $M_1$  and
$M_3$ is a saddle and another one is a node). Therefore
considering \eqref{signs:conf-3.4} we obtain the relations:
$$
\operatorname{sign}(\Delta_1)=-\operatorname{sign}(\Delta_3)=\operatorname{sign}(B_3U_1);\quad
\operatorname{sign}(\widetilde{\Delta}_3)=-\operatorname{sign}(\widetilde{\Delta}_1)=\operatorname{sign}(U_2B_3U_1).
$$
Assuming that the conditions on the right side of
\eqref{cond:portraits-3.4} are satisfied  and considering the
equalities above it is easy to convince ourselves that  we get
univocally the corresponding phase portrait, except in the case
{Picture 3.4($\stackrel{\,*}{h}$4)}.  For this case it is
necessary to consider the stability of the focus  $M_4$ and of the
node $M_3$   of systems \eqref{sys:Config.3.4-G}.  Calculations
yield: $\rho_3\rho_4=\frac{f+h-2fh}{1-h}$.   Therefore in the case
$h<0$ and $f<0$ we obtain $\rho_3\rho_4<0$, i.e. these
singularities have different stabilities.  This completes the
proof of the claim.

\subsubsection{The phase portraits associated with {Config.
3.5}} According to Table 2 we shall consider the family of systems
\begin{equation}\label{sys:Config.3.5-G}
\dot x = x [1  + (1-h)(x-y)], \quad \dot y  = y(f- hx+hy),
\end{equation}
for which the condition
\begin{equation}\label{cond:A3-A}
h(h-1)f(f-1)(f+h-fh)\ne0
\end{equation}
holds. We observe that this family of systems is a particular case
of the family  \eqref{sys:Config.3.1-G} when we have $g=1-h$ and
in this case the singular point $M_4$ has coalesced  with $R_1(1,
1, 0)$.  So these systems    possess three simple finite
singularities and three  infinite singularities (one of which
being double).   For   the singularities  of these systems with
the condition \eqref{cond:A3} we have for the finite
singularities:
\begin{equation}\label{expr:Delta-i,delta-i:3.5}
\begin{gathered}
M_1(0,0):\  \Delta_1=f,  \ \   \delta_1=(1-f)^2;\\
M_2(1/(h-1),0):\   \Delta_2= (fh-f-h)/(1-h), \ \ \delta_3= {(fh-f-1)^2}/{(h-1)^2};\\
M_3(0,-f/h):\    \Delta_3=f(fh-f-h)/h, \ \ \delta_3= {(f + h)^2}/{h^2}
\end{gathered}
\end{equation}
 and for the infinite ones
\begin{gather}\label{ISPs:tilde delta-i:3.5}
 R_1(1, 1, 0):\  \widetilde{\Delta}_1= 0, \  \widetilde \rho_1=-1;
  R_2(1, 0, 0)  :\  \widetilde{\Delta}_2 = 1-h;\ \
   R_3(0, 1, 0) :\  \widetilde{\Delta}_3 = h.
\end{gather}
Since $\widetilde \rho_1\ne0$ the singular point $R_1(1, 1, 0)$ is
a saddle-node (both hyperbolic sectors being on the same part of
the infinite line).


Evaluating  the  invariant polynomials we need for systems
\eqref{sys:Config.3.5-G}  we obtain:
\begin{equation}\label{val:conf-3.5}
\begin{gathered}
\mu_0= 0,\quad K=  2h(h-1)(x-y)^2 =-2\widetilde{\Delta}_2\widetilde{\Delta}_3(x-y)^2,\\
 B_3= 3(f-1)(fh-f-h)x^2y^2=3(f-1) \Delta_2 \widetilde{\Delta}_2x^2y^2,\\
 U_{1}= \frac{1}{8}f(f-1)(fh-f-h)h^2(1 - h)^2 =
   \frac{1}{8}(f-1)\Delta_1\Delta_2 \widetilde{\Delta}_2^3\widetilde{\Delta}_3^2,\\
 U_{3}= \frac{1}{2}(f+1)h(h-1)(fh-f-h) =
 -\frac{1}{2}(f+1)\Delta_2\widetilde{\Delta}_2^2\widetilde{\Delta}_3.\\
\end{gathered}
\end{equation}
 Herein considering \eqref{expr:Delta-i,delta-i:3.5} and
\eqref{ISPs:tilde delta-i:3.5} we get the following relations:
\begin{equation}\label{signs:conf-3.5}
\operatorname{sign}(B_3U_1) = \operatorname{sign}(\Delta_1);\quad \text{if}\ \
     B_3U_1 >0\  \Rightarrow\
  \operatorname{sign}(U_3)=  -\operatorname{sign}( \Delta_2 \widetilde{\Delta}_3).
\end{equation}
We observe that  systems  \eqref{sys:Config.3.5-G}  possess three
finite simple singularities. Therefore   considering the fact that
these systems possess neither   a focus nor a center, according to
\cite{ArtLliVul08-IJBCh} (see Table 1)    we obtain two saddles
and one node if $K<0$, and one saddle and two nodes if $K>0$.

On the other hand by \eqref{val:conf-3.5} we have
$\operatorname{sign}(K)
=-\operatorname{sign}(\widetilde{\Delta}_2\widetilde{\Delta}_3)$. So
clearly  besides the saddle-node $R_1(1,1,0)$  systems
\eqref{sys:Config.3.5-G} possess at the infinity   two nodes if
$K<0$ and they have a node and a saddle if $K>0$.

\begin{remark}\label{rem:N2-node}{\rm
Without loss of generality we  assume that the infinite
singular point $R_2(1,0,0)$ is a node due to the substitution
$(x,y,t,f,h)\mapsto(x/f,y/f,ft,1/f,1-h)$, which leads us to  the
systems \eqref{sys:Config.3.5-G} but interchanges  the points
$R_2$ and $R_3$.}
\end{remark}

\paragraph{The case $K<0$.} Then  besides
the saddle-node $R_1(1,1,0)$,  systems \eqref{sys:Config.3.5-G}
possess at the infinity   two nodes.

\subparagraph{\it The subcase $B_3U_1<0$.}  Then $ {\Delta}_1 <0$
(i.e. $M_1$ is a saddle) and hence one of the singular points
$M_2$ or $M_3$ is a saddle and another one is a node. Therefore we
have either $\Delta_2>0$ and $\Delta_3<0$ or $\Delta_2<0$ and
$\Delta_3>0$. Considering the relations $f>0$ and $0<h<1$ (which
fix the position of the singularities) as well as the two nodes
and the saddle-node at infinity in both   cases we get the phase
portraits topologically equivalent to {Picture 3.5(g1)}.

\subparagraph{The  subcase $ B_3U_1>0$.}  In this case we have
 ${\Delta}_1 >0$ (i.e. $f>0$) and  $M_1$ is a node. Then the
remaining two finite singularities are saddles and this leads to
{Picture 3.5(g2)}.

\paragraph{The case $K>0$.} Systems
\eqref{sys:Config.3.5-G} possess as finite singularities a saddle
and two nodes and at infinity they have   a saddle and a node
besides the saddle-node. According to Remark \ref{rem:N2-node} we
may assume that $R_2$ is a node and $R_3$ is a saddle (i.e.
$h<0$).

\subparagraph{The  case $ B_3U_1<0$.}  We have $
{\Delta}_1 <0$ (i.e. $f<0$) and therefore $M_1$ is a saddle
whereas the remaining points are nodes. Considering the saddle
$R_3$ and the node $R_2$   we obtain the phase portrait given by
{Picture 3.5(h1)}.

\subparagraph{The  case $ B_3U_1>0$.}  In this case we obtain $
{\Delta}_1 >0$ (i.e. $f>0$) and hence $M_1$ is a node. Since
$\widetilde{\Delta}_3<0$ (as $R_3$ is a saddle) considering
\eqref{signs:conf-3.5} we obtain $\operatorname{sign}(U_3)= \operatorname{sign}({\Delta}_2)$.

 {\it I. The  subcase $U_3<0$.} Then ${\Delta}_2<0$,
i.e. $M_2$ is a saddle and $M_3$ is a node. This leads univocally
to the phase portrait given {Picture 3.5(h2)}.

 {\it II. The  subcase $U_3>0$.}  In this case $M_2$
is a node and $M_3$  is a saddle and considering the location of
all the singularities we get a phase portrait which is
topologically equivalent to {Picture 3.5(h1)}.


\subsubsection{The phase portraits associated with {Config. 3.6}} 
According to Table 2 we   consider the one-parameter family
of systems
\begin{equation}\label{sys:Config.3.6-G}
\dot x = x [1  + (h-1)y ], \quad \dot y  = y(-x+hy), \quad h(h -1) \ne0.
\end{equation}
We observe that this family of systems is a particular case of the
family  \eqref{sys:Config.3.2-G} when $g=0$. Hence in this case we
have two pairs of  singularities such that in each pair the two
singularities have coalesced: $M_3$ with $M_1$ and $M_2$ with
infinite point $R_2(1, 0, 0)$.   So for the singularities  of
these systems we have
\begin{equation}\label{expr:Delta-i,delta-i:3.6}
\begin{aligned}
M_3\equiv M_1(0,0):\ &  \Delta_1=0,  \ \ \rho_1=1;\\
M_4\Big(\frac{h}{1-h},\frac{1}{1-h}\Big):\ & \Delta_4=
 \frac{h}{h-1}=-\rho_4, \ \ \delta_4=  \frac{h(4-3h)} {(h-1)^2}
\end{aligned}
\end{equation}
 for the finite singularities and
\begin{gather}\label{ISPs:tilde delta-i:3.6}
 R_1(1, 1, 0):\  \widetilde{\Delta}_1= 1-h;\ \
  R_2(1, 0, 0)  :\  \widetilde{\Delta}_2 = 0, \  \widetilde \rho_2= 1;\ \
   R_3(0, 1, 0) :\  \widetilde{\Delta}_3 = h
\end{gather}
for the infinite ones. Clearly both double points $M_1(0,0)$ and
$R_2(1, 0, 0)$ are saddle-nodes. Moreover, for the second point
both hyperbolic sectors are on the same part of the infinite line.

For systems \eqref{sys:Config.3.6-G} we have:
\begin{equation}\label{val:conf-3.6}
\begin{gathered}
K= 2h(h-1)y^2= 2\Delta_4\widetilde{\Delta}_1^2y^2 =-2\widetilde{\Delta}_1\widetilde{\Delta}_3y^2,\\
 H_1= 288h=288\widetilde{\Delta}_3,\\
W_4= h^3(4-3h)=\delta_4\widetilde{\Delta}_1^2\widetilde{\Delta}_3^2.\\
\end{gathered}
\end{equation}
Herein we observe that the invariant polynomials above govern the
types of the simple singular points of systems
\eqref{sys:Config.3.6-G}. More precisely the types of the
singularities $M_4$, $R_1$ and $R_3$ are determined by the
following conditions, respectively:
$$
\begin{array}{cll}
 \text{(i)}  & K<0 &  \Rightarrow\quad M_4 - \text{saddle}, R_1, R_3 -  \text{nodes};\\
 \text{(ii)} & K>0, W_4<0, H_1<0 &  \Rightarrow\quad M_4 - \text{focus}, R_1 - \text{node}, R_3 - \text{saddle};\\
 \text{(iii)} & K>0, W_4\ge0  &  \Rightarrow\quad M_4 - \text{node}, R_1 - \text{saddle}, R_3 -  \text{node};\\
 \text{(iv)} & K>0, W_4<0,  H_1>0   &  \Rightarrow\quad M_4 - \text{focus}, R_1 - \text{saddle}, R_3 -  \text{node}.\\
\end{array}
$$
Then considering the location of the singular point $M_4$ in each
one of the cases above we arrive at a phase portrait  given by:
{Picture 3.6(k1)}  in the case  \text{(i)};
{Picture 3.6($\stackrel{\,*}{l}$1)}  in the case
\text{(ii)};  {Picture 3.6(l2)}  in the case
\text{(iii)}  and {Picture 3.6($\stackrel{\,*}{l}$2)}  in
the case \text{(iv)}.

We stress that in the case of {Picture
3.6($\stackrel{\,*}{l}$1)} the behaviour of the trajectories in
the vicinity of the focus $M_4$ is determined univocally due to
the relation $\rho_1\rho_4=-\Delta_4<0$ and this means that the
stability of the focus is opposite to the stability of the
parabolic sector  of the saddle-node $M_1$.

\subsubsection{The phase portraits associated with {Config.
3.7}}
According to Table 2 a system possessing this configuration
belongs to  the one-parameter family of systems
\begin{equation}\label{sys:Config.3.7-G}
\dot x = x [h-1  + (h-1)y ], \quad \dot y  = y(h-x+hy), \quad
h(h - 1) \ne0.
\end{equation}
Comparing the singularities of these systems with those of the
systems \eqref{sys:Config.3.1-G} we observe that in this case we
have two pairs of  singularities such that in each pair the two
singularities have coalesced: $M_4$ with $M_3$ and  $M_2$ with the
infinite point $R_2(1, 0, 0)$. So  for  the singularities  of
these systems we obtain:
\begin{equation}\label{expr:Delta-i,delta-i:3.7}
M_1(0,0):  \Delta_1=h(h-1);\quad M_4\equiv M_3(0,-1):\
\Delta_3=0, \ \rho_3=-h
\end{equation}
for the finite singularities, and
\begin{equation}\label{ISPs:tilde delta-i:3.7}
 R_1(1, 1, 0):\  \widetilde{\Delta}_1= 1-h;\ \
  R_2(1, 0, 0)  :\  \widetilde{\Delta}_2 = 0, \  \widetilde \rho_2= 1;\ \
   R_3(0, 1, 0) :\  \widetilde{\Delta}_3 = h
\end{equation}
for the infinite ones. Clearly both double points $M_3(0,0)$ and
$R_2(1, 0, 0)$ are saddle-nodes and  the  hyperbolic sectors of
the second saddle-node are located on the same part of the
infinite line.

For systems \eqref{sys:Config.3.7-G} we calculate:
\begin{equation}\label{val:conf-3.7}
\begin{gathered}
K=  2h(h-1)y^2= 2\Delta_1y^2\quad \Rightarrow\quad
\operatorname{sign}(K)=\operatorname{sign}(\Delta_1)=
  -\operatorname{sign}(\widetilde{\Delta}_1\widetilde{\Delta}_3);\\
 H_5=  384h^2(1-h)^3=384\widetilde{\Delta}_1^3\widetilde{\Delta}_3^2\quad
 \Rightarrow\quad \operatorname{sign}(H_5)=\operatorname{sign}(\widetilde{\Delta}_1).
\end{gathered}
\end{equation}
Herein we observe that the invariant polynomials $K$ and $H_5$
govern the types of the simple singular points of systems
\eqref{sys:Config.3.7-G}. More precisely the types of the
singularities $M_1$, $R_1$ and $R_3$ are determined by the
following conditions, respectively:
$$
\begin{array}{cll}
 \text{(i)}  & K<0 &  \Rightarrow\quad M_1 - \text{saddle}, R_1, R_3 -  \text{nodes};\\
 \text{(ii)} & K>0,  H_5<0 &  \Rightarrow\quad M_1 - \text{node}, R_1 - \text{saddle}, R_3 - \text{node};\\
 \text{(iii)} & K>0,  H_5>0 &  \Rightarrow\quad M_1 - \text{node}, R_1
 - \text{node}, R_3 -  \text{saddle}.
\end{array}
$$
Since the coordinates of all  positions of the singularities are
determined, in each one of the cases above we arrive univocally at
the phase portrait given by: {Picture 3.7(k1)} in the case
\text{(i)}; {Picture 3.7(l2)}  in the case \text{(ii)}
and {Picture 3.7(l1)}  in the case \text{(iii)}.

\subsubsection{The phase portraits associated with {Config. 3.8}}
According to Table 2 a system possessing this configuration
belongs to  the one-parameter family of systems
\begin{equation}\label{sys:Config.3.8-G}
\dot x = x [1  + (1-h)(x-y)], \quad \dot y  = hy(y- x), \quad
h(h - 1) \ne0.
\end{equation}
Comparing the singularities of these systems with those of the
systems \eqref{sys:Config.3.1-G} we observe that in this case we
have again two pairs of  singularities such that in each pair the
two singularities have coalesced: $M_3$ with $M_1$ and  $M_4$ with
the infinite point $R_1(1, 1, 0)$.    So  for  the singularities
of these systems we obtain:
\begin{equation}\label{expr:Delta-i,delta-i:3.8}
M_3\equiv M_1(0,0):\  \Delta_1=0, \ \rho_1=1;
 \quad   M_2(1/(h-1),0):\ \Delta_2=h/(h-1)
\end{equation}
for the finite singularities and
\begin{equation}\label{ISPs:tilde delta-i:3.8}
  R_1(1, 1, 0)  :\  \widetilde{\Delta}_1 = 0, \  \widetilde \rho_1= -1;\ \
 R_2(1, 0, 0):\  \widetilde{\Delta}_2= 1-h;\ \
   R_3(0, 1, 0) :\  \widetilde{\Delta}_3 = h
\end{equation}
for the infinite ones. Clearly   both double points $M_1(0,0)$ and
$R_1(1, 0, 0)$ are saddle-nodes and  the  hyperbolic sectors of
the second saddle-node are located on the same part of the
infinite line.

For systems \eqref{sys:Config.3.8-G} we calculate:
\begin{equation}\label{val:conf-3.8}
\begin{gathered}
K=  2h(h-1)(x-y)^2= 2\Delta_1\widetilde{\Delta}_2^2(x-y)^2\\
\Rightarrow\  \operatorname{sign}(K)=\operatorname{sign}(\Delta_1)=
  -\operatorname{sign}(\widetilde{\Delta}_2\widetilde{\Delta}_3);\\
H_5= 384h^3=384\widetilde{\Delta}_3^3\
  \Rightarrow\  \operatorname{sign}(H_5)=\operatorname{sign}(\widetilde{\Delta}_3).
\end{gathered}
\end{equation}
So we again obtain that these invariant polynomials determine
completely the types of the simple singularities. Thus applying
the same arguments as above  we get for the systems
\eqref{sys:Config.3.8-G} the following phase portraits:
{Picture 3.8(k1)} if $K<0$; {Picture 3.8(l1)} if
$K>0$ and $H_5<0$; and {Picture 3.8(l2)} if $K>0$ and
$H_5>0$.


\subsubsection{The phase portraits associated with {Config. 3.9}}
According to Table 2 we   consider the family of systems
\begin{equation}\label{sys:Config.3.9-G}
\dot x =x(1+ gx+ y), \quad \dot y  = y(f-x+gx+y),
\end{equation}
for which the condition
\begin{equation}\label{cond:A4}
g(g-1)f(f-1)(1-g+fg)\ne0
\end{equation}
 holds.
For the all four distinct finite singularities of systems
\eqref{sys:Config.3.9-G} with the condition
\eqref{cond:A4} we have
\begin{equation}\label{expr:Delta-i,delta-i:3.9}
\begin{gathered}
M_1(0,0):\quad \Delta_1=f, \quad \rho_1=f+1,\quad  \delta_1=(f-1)^2;\\
M_2(-1/g,0):\quad \Delta_2=(g-1-fg)/g, \quad \delta_2=(1+fg)^2/g^2;\\
M_3(0,-f):\quad \Delta_3=f(f-1), \quad \delta_3=1-2f;\\
M_4(f-1,  g-1-fg):\\
 \Delta_4= (f-1)(g-1-fg), \rho_4=-1,\;
 \delta_4=4g(f-1)^2+4f-3.
\end{gathered}
\end{equation}
 and for the two infinite singular points we obtain
\begin{equation}\label{ISPs:tilde delta-i:3.9}
  R_2(1, 0, 0): \  \widetilde{\Delta}_2 = g;\quad
 R_1\equiv R_3(0, 1, 0): \ \widetilde{\Delta}_3 = 0, \widetilde \rho_3=1.
\end{equation}
 We note that in this case the infinite singularity $R_3(0, 1, 0)$
is a  saddle-node for which the   infinite line serves as a
separatrix for the hyperbolic sectors.

Taking into consideration  \eqref{expr:Delta-i,delta-i:3.9} and
\eqref{ISPs:tilde delta-i:3.9} we evaluate for systems
\eqref{sys:Config.3.9-G} the invariant polynomials we need:
\begin{equation}\label{val:conf-3.9}
\begin{gathered}
\mu_0= g=  \widetilde{\Delta}_2, \ B_3=  3(1-f)x^2y^2,\\
K= 2 g(g-1) x^2 + 4 g x y + 2 y^2,\\
 U_{1}= \frac{1}{8} f(1-f)(g-1)^2=\frac{1}{8}\Delta_1(1-f)(g-1)^2,\\
U_{4}=  f(1-f)^2(g-1-fg)= \Delta_1 {\Delta}_2\widetilde{\Delta}_2(f-1)^2,\\
H_4=  48 (1-f)=-48{\Delta}_3/\Delta_1.
\end{gathered}
\end{equation}
Herein considering the condition \eqref{cond:A4} we evidently
obtain the relations
\begin{equation}\label{sign:Conf-3.9}
\begin{gathered}
\operatorname{sign}(\mu_0)=\operatorname{sign}(\widetilde{\Delta}_2),\quad
\operatorname{sign}(B_3U_1)=\operatorname{sign}({\Delta}_1),\\
\operatorname{sign}(U_4)=\operatorname{sign}(\Delta_1 {\Delta}_2\widetilde{\Delta}_2),\quad
\operatorname{sign}(H_4)=-\operatorname{sign}(\Delta_1\Delta_3).\\
\end{gathered}
\end{equation}

 \paragraph{The case $\mu_0<0$.}
 As $\mu_0=\operatorname{Discrim}(K)/16$ by \eqref{val:conf-3.9} we conclude that
$K>0$. Therefore  according to \cite{ArtLliVul08-IJBCh} (see Table
1) on the finite part of the phase plane  systems
\eqref{sys:Config.3.9-G} possess one saddle  and three
anti-saddles. Since three singularities are on the invariant
lines, clearly only one anti-saddle could be a focus. Considering
\cite{ArtLliVul08-IJBCh} (see Table 1), apart from the saddle we
have three nodes if either $W_4>0$ or $W_4=0$ and $W_3\ge0$; and
we have two nodes and a focus  if either $W_4<0$ or $W_4=0$ and
$W_3<0$.

On the other hand,  by \eqref{sign:Conf-3.9} we get
$\widetilde{\Delta}_2<0$, i.e. the infinite singularity
$R_2(1,0,0)$ is a saddle.

\begin{remark}\label{rem:M1-no-sdl} \rm
We note that in the case $\mu_0<0$ and $\delta_4\ge0$ (i.e. when
$M_4$ is a node)  the singular point $M_1$ should be a node.
\end{remark}
Indeed  suppose  that $M_1$ is a saddle. Considering
\eqref{expr:Delta-i,delta-i:3.9} the conditions $g<0$ and $f<0$
imply $\delta_4<0$, i.e. we get a contradiction.

Herein considering \eqref{sign:Conf-3.9}, we obtain that the types
of the finite singularities of systems \eqref{sys:Config.3.9-G}
are determined by the following conditions, respectively:
$$
\begin{array}{cllcccc}
   &   &   &  M_1  & M_2 & M_3 &  M_4  \\
\text{(i)}  & (W_4>0)\vee(W_4=0,W_3\ge0),U_4<0, H_4<0 &  
 \Rightarrow & \tilde n & \tilde n & \tilde n & \tilde s ;\\
\text{(ii)}  & (W_4>0)\vee(W_4=0,W_3\ge0),U_4<0, H_4>0 &  
 \Rightarrow &  \tilde n & \tilde n & \tilde s & \tilde n ;\\
\text{(iii)}  & (W_4>0)\vee(W_4=0,W_3\ge0),U_4>0 &  
 \Rightarrow &  \tilde n & \tilde s & \tilde n & \tilde n ;\\
\text{(iv)}  & (W_4<0)\vee(W_4=0,W_3<0),B_3U_1>0, U_4<0 &  
 \Rightarrow &  \tilde n & \tilde n & \tilde s & \tilde f ;\\
\text{(v)}  & (W_4<0)\vee(W_4=0,W_3<0),B_3U_1>0, U_4>0 &  
 \Rightarrow &  \tilde n & \tilde s & \tilde n & \tilde f ;\\
\text{(vi)}  & (W_4<0)\vee(W_4=0,W_3<0),B_3U_1<0 & 
 \Rightarrow &  \tilde s & \tilde n & \tilde n & \tilde f.
\end{array}
$$
We note that in the   case \text{(vi)} we have $\rho_3\rho_4<0$,
i.e. the node $M_3$ and the focus $M_4$ are of the opposite
stabilities.  So considering the infinite singularities $R_2(1, 0,
0)$ (a saddle) and $R_3(0, 1, 0)$ (a saddle-node) we arrive in
each of the mentioned cases to the following phase portrait,
respectively:
\begin{gather*}
\text{(i)\ 3.9(b1)};\quad
\text{(ii)\ 3.9(b2)};\quad
\text{(iii)\  3.9(b3)};\\
\text{(iv)\  3.9($\stackrel{\,*}{b}$2)};\quad
\text{(v)\  3.9($\stackrel{\,*}{b}$3)};\quad
\text{(vi)\ 3.9($\stackrel{\,*}{b}$4)}.
\end{gather*}

\paragraph{The case $\mu_0>0$.}
 According to \cite{ArtLliVul08-IJBCh}
(see Table 1) on the finite part of the phase plane,   systems
\eqref{sys:Config.3.9-G} possess two saddles and two anti-saddles.
Moreover as only one anti-saddle could be a focus, besides the
saddles we have two nodes if  $W_4\ge0$   and we have a node and a
focus if  $W_4<0$.

On the other hand,  by \eqref{sign:Conf-3.9} we get
$\widetilde{\Delta}_2>0$, i.e. the infinite singularity
$R_2(1,0,0)$ is a node.

Thus considering \eqref{sign:Conf-3.9} we obtain that the types of
the finite singularities of systems \eqref{sys:Config.3.9-G} are
determined by the following conditions, respectively:
$$
\begin{array}{cllcccc}
   &   &   &  M_1  & M_2 & M_3 &  M_4  \\
\text{(i)}  &  W_4\ge0,B_3U_1<0,U_4<0  &  \Rightarrow &  \tilde s & \tilde n & \tilde n & \tilde s ;\\
\text{(ii)}  & W_4\ge0,B_3U_1<0,U_4>0 &  \Rightarrow & \tilde s & \tilde s & \tilde n & \tilde n ;\\
\text{(iii)}  & W_4\ge0,B_3U_1>0,H_4<0  &  \Rightarrow &  \tilde n & \tilde s & \tilde n & \tilde s ;\\
\text{(iv)}  &  W_4\ge0,B_3U_1>0,H_4>0 , U_4<0 &  \Rightarrow &  \tilde n & \tilde s & \tilde s & \tilde n ;\\
\text{(v)}  & W_4\ge0,B_3U_1>0,H_4>0 , U_4>0 &  \Rightarrow &  \tilde n & \tilde n & \tilde s & \tilde s ;\\
\text{(vi)}  &  W_4<0,B_3U_1>0 &  \Rightarrow &  \tilde n & \tilde s & \tilde s & \tilde f;\\
\text{(vii)}  &  W_4<0,B_3U_1<0 &  \Rightarrow &  \tilde s & \tilde s & \tilde n & \tilde f .\\
\end{array}
$$
We note that  in cases \text{(i)}, \text{(iv)} and
\text{(v)} we obtain  phase portraits which are topologically
equivalent to the same portrait, given by {Picture
3.9(c1)}. Therefore this picture occurs if and only if $W_4\ge0$
and either $B_3U_1<0$ and $U_4<0$, or $B_3U_1>0$ and $H_4>0$.

Examining all the cases above considering the infinite
singularities $R_2(1, 0, 0)$ (a node) and $R_3(0, 1, 0)$ (a
saddle-node) we arrive in each of the remaining cases   to one of
the   phase portraits:
$$
\text{(ii)\ 3.9(c2)};\quad \text{(iii)\ 3.9(c3)};\quad
\text{(vi)\ 3.9($\stackrel{\ *}{c}$1)};\quad
\text{(vii)\ 3.9($\stackrel{\ *}{c}$2)} .
$$
Thus we arrive exactly at the respective conditions  given by
Table 5 in this case.

\subsubsection{The phase portraits associated with {Config. 3.10}}
According to Table 2 a system possessing this configuration
belongs to  the one-parameter family of systems
\begin{equation}\label{sys:Config.3.10-G}
\dot x = x(g+ gx+ y), \quad \dot y  = y[g-1+(g-1)x+y], \quad
g(g-1)\ne0.
\end{equation}
These systems possess three finite singularities (one of them
being double) and two infinite (one double). For the  finite
singularities of systems
\eqref{sys:Config.3.10-G}   we have
\begin{equation}\label{expr:Delta-i,delta-i:3.10}
\begin{gathered}
M_1(0,0): \  \Delta_1=g(g-1);\quad M_3(0,1-g): \  \Delta_3=1-g;\\
M_4\equiv M_2(-1,0): \quad \Delta_2=0, \quad \rho_2=-g \\
\end{gathered}
\end{equation}
and for the two infinite singular points we obtain
\begin{equation}\label{ISPs:tilde delta-i:3.10}
  R_2(1, 0, 0): \  \widetilde{\Delta}_2 = g;\quad
 R_1\equiv R_3(0, 1, 0): \ \widetilde{\Delta}_3 = 0, \widetilde \rho_3=1.
\end{equation}
For systems
\eqref{sys:Config.3.10-G} calculations yield:
\begin{equation}\label{val:conf-3.10}
\mu_0= g, \ B_3=  3gx^2y^2, \quad
 U_{1}= \frac{1}{8}g^2(g-1)^3.
\end{equation}
 Herein considering \eqref{expr:Delta-i,delta-i:3.10} and
\eqref{ISPs:tilde delta-i:3.10} we obtain the following relations:
\begin{equation}\label{sign:Conf-3.10}
\operatorname{sign}(\mu_0)=\operatorname{sign}(\widetilde{\Delta}_2)
=-\operatorname{sign}({\Delta}_1{\Delta}_3),\quad
\operatorname{sign}(B_3U_1)=\operatorname{sign}({\Delta}_1).
\end{equation}
So we observe that the two invariant polynomials $\mu_0$ and
$B_3U_1$ determine completely the types of the simple
singularities. More exactly we obtain that the types of all the
singularities of systems \eqref{sys:Config.3.10-G} (for infinite
points we denote them by capital letters) and they are determined
by the following conditions, respectively:
$$
\begin{array}{cllccccc}
   &   &   &  M_1  & M_2 & M_3 &  R_2 & R_3  \\
\text{(i)}  &  \mu_0<0  &  \Rightarrow &  \tilde n &  \tilde s\tilde n  & \tilde n & \widetilde S &  \widetilde S\widetilde N ;\\
\text{(ii)}  &  \mu_0>0,B_3U_1<0  &  \Rightarrow &  \tilde s &  \tilde s\tilde n  & \tilde n & \widetilde N &  \widetilde S\widetilde N ;\\
\text{(iii)}  &  \mu_0>0,B_3U_1>0  &  \Rightarrow &  \tilde n &  \tilde s\tilde n & \tilde s & \widetilde N & \widetilde S\widetilde N .\\
\end{array}
$$
These types of   singularities univocally lead to the following
phase portraits, respectively:
$$
\text{(i)}\text{ Picture 3.10(e1)};\quad
\text{(ii)}\text{ Picture 3.10(f1)};\quad \text{Picture 3.10(f2)}.
$$

\subsubsection{The phase portraits associated with {Config. 3.11}}
According to Table 2 a system possessing this configuration
belongs to  the one-parameter family of systems
\begin{equation}\label{sys:Config.3.11-G}
\dot x = x(1+ gx+ y), \quad \dot y  = y(-x+gx+y), \quad
g(g-1)\ne0.
\end{equation} These systems possess the following finite singularities:
\begin{equation}\label{expr:Delta-i,delta-i:3.11}
\begin{gathered}
M_3\equiv M_1(0,0): \  \Delta_1=0, \rho_1=1;\ \
 M_2(-1/g,0): \  \Delta_2=\frac{g-1}{g}, \rho_2=\frac{1-2g}{g};\\
 M_4(-1,g-1): \quad \Delta_4=1-g, \ \rho_4=-1, \ \delta_4=4g-3
\end{gathered}
\end{equation}
and  infinite ones:
\begin{equation}\label{ISPs:tilde delta-i:3.11}
  R_2(1, 0, 0): \  \widetilde{\Delta}_2 = g;\quad
 R_1\equiv R_3(0, 1, 0): \ \widetilde{\Delta}_3 = 0, \widetilde \rho_3=1.
\end{equation}
For systems \eqref{sys:Config.3.11-G} calculations yield:
\begin{equation}\label{val:conf-3.11}
\mu_0= g, \quad W_4=  4g-3, \quad
 H_5= \ 384(1-g).
\end{equation}
 Herein we obtain:
\begin{equation}\label{sign:Conf-3.11}
\operatorname{sign}(\mu_0)=\operatorname{sign}(\widetilde{\Delta}_2)=-\operatorname{sign}({\Delta}_2{\Delta}_4),
\ \operatorname{sign}(H_5)=\operatorname{sign}({\Delta}_4), \ \operatorname{sign}(W_4)=\operatorname{sign}({\delta}_4).
\end{equation}
So we obtain that the types of all the singularities of systems
\eqref{sys:Config.3.11-G}  are determined by the following
conditions, respectively:
$$
\begin{array}{cllccccc}
   &   &   &  M_1  & M_2 & M_4 &  R_2 & R_3  \\
\text{(i)}  &  \mu_0<0  &  \Rightarrow &  \tilde s\tilde n  & \tilde n & \tilde f & \widetilde S &  {\widetilde S\widetilde N};\\
\text{(ii)}  &  \mu_0>0,W_4\ge0,H_5>0  &  \Rightarrow &   \tilde s\tilde n  & \tilde s & \tilde n & \widetilde N &  {\widetilde S\widetilde N};\\
\text{(iii)}  &  \mu_0>0, W_4<0  &  \Rightarrow &   \tilde s\tilde n  & \tilde s & \tilde f & \widetilde N &  {\widetilde S\widetilde N};\\
\text{(iv)}  &  \mu_0>0,W_4\ge0, H_5<0  &  \Rightarrow &   \tilde s\tilde n  & \tilde n & \tilde s & \widetilde N &  {\widetilde S\widetilde N}.\\
\end{array}
$$
We observe that in the  case $\mu_0<0$ (i.e. $g<0$) the condition
$\rho_2\rho_4=(2g-1)/g>0$, i.e. the stabilities of the node $M_2$
and of the focus $M_4$ coincide. So considering the types of the
singular points above we get univocally   the following phase
portraits, respectively:
\begin{gather*}
\text{(i)}\text{ Picture 3.11($\stackrel{\ *}{e}$1)};\quad
\text{(ii)}\text{ Picture 3.11(f1)};\\\
\text{(iii)}\text{ Picture 3.11($\stackrel{\ *}{f}$1)};\quad \text{(iv)}
\text{ Picture 3.11(f2)}.
\end{gather*}

\subsubsection{The phase portraits associated with {Config.
3.12}}
 According to Table 2 we   consider the family of systems
\begin{equation}\label{sys:Config.3.12-G}
\dot x = x(1+  y), \quad \dot y  = y(f+x+y),\quad f(f-1)\ne0
\end{equation}
 which possess the following five singularities:
\begin{equation}\label{expr:Delta-i,delta-i:3.12}
\begin{gathered}
M_1(0,0):\quad \Delta_1=f,\quad \rho_1=f+1;\\
M_3(0,-f):\quad \Delta_3=f(f-1),\quad \rho_3=1-2f;\\
M_4\left(1-f,  -1\right):\quad \Delta_4=1-f,\quad
\rho_4=-1,\quad \delta_4=4f-3
\end{gathered}
\end{equation}
and
\begin{equation}\label{ISPs:tilde delta-i:3.12}
 R_1=R_3(0, 1, 0):\quad \widetilde{\Delta}_3 =0, \widetilde \rho_3=1;\quad
  R_2(1, 0, 0):\quad  \widetilde{\Delta}_2 = 0, \widetilde \rho_2=-1.
\end{equation}

\begin{remark}\label{rem:two-S-N}
{\rm We observe that both infinite points are double  and they are
saddle-nodes. However for   $R_3(0, 1, 0)$     the infinite line
serves as a separatrix for the hyperbolic sectors, whereas both
hyperbolic sectors of the  saddle-node $R_2(1, 0, 0)$  are located
on the same part of the infinite line.}
\end{remark}


Considering  \eqref{expr:Delta-i,delta-i:3.12} for systems
\eqref{sys:Config.3.12-G}  we calculate
\begin{equation}\label{val:conf-3.12}
\begin{gathered}
B_3= 3(f-1) x^2y^2=-3{\Delta}_4x^2y^2, \
 U_{1}=  \frac{1}{8}f  (f-1)=-\frac{1}{8}\Delta_1{\Delta}_4,\\
 H_5=  384 (1-f)=384 {\Delta}_4, \
 W_4= (f-1)^2 (4f-3)= {\Delta}_4^2{\delta}_4.\\
\end{gathered}
\end{equation}
Herein we obtain
\begin{equation}\label{signs:conf-3.12}
    \operatorname{sign}(B_3U_1) = \operatorname{sign}(\Delta_1);\quad
  \operatorname{sign}(H_5)=  \operatorname{sign}({\Delta}_4);\quad   \operatorname{sign}(W_4)=  \operatorname{sign}({\delta}_4).
\end{equation}
 So these invariant polynomials determine the types of all the
finite singularities of systems
\eqref{sys:Config.3.12-G}  as follows:
$$
\begin{array}{cllccc }
   &   &   &  M_1  & M_3 & M_4    \\
\text{(i)}  &  B_3U_1<0  &  \Rightarrow &  \tilde s & \tilde n & \tilde f ;\\
\text{(ii)}  &  B_3U_1>0, H_5>0, W_4\ge0  &  \Rightarrow &  \tilde n & \tilde s & \tilde n;\\
\text{(iii)}  &  B_3U_1>0, H_5>0, W_4<0  &  \Rightarrow &  \tilde n & \tilde s & \tilde f;\\
\text{(iv)}  &  B_3U_1>0, H_5<0  &  \Rightarrow &  \tilde n & \tilde n & \tilde s.\\
\end{array}
$$
We observe that in the  case $B_3U_1<0 $ (i.e. $f<0$) the
condition $\rho_3\rho_4= 2f-1<0$, i.e. the stabilities of the node
$M_3$ and of the focus $M_4$ are opposite. So considering the
types of the singular points above and Remark \ref{rem:two-S-N} we
get univocally   the following phase portraits, respectively:
\begin{gather*}
\text{(i)}\text{ Picture 3.12($\stackrel{\,*}{h}$1)};\quad
\text{(ii)}\text{ Picture 3.12(h2)};\\
\text{(iii)}\text{ Picture 3.12($\stackrel{\,*}{h}$2)};\quad \text{(iv)}
\text{ Picture 3.12(h3)}.
\end{gather*}
As we have the one-parameter family of systems the conditions
above could be simplified. More precisely as the bifurcation value
$f=3/4$ (respectively $f=0$; $f=1$)  for the parameter $f$ is
given by polynomial  $W_4$ (respectively $B_3U_1$; $H_5$),  we
get, for the respective phase portraits, the  conditions given by
Table~5.

\subsubsection{The phase portraits associated with {Config. 3.13}}

According to Table 2 this configuration corresponds to the
normal form
\begin{equation}\label{sys:Config.3.13-G}
\dot x = x(1+  y), \quad \dot y  = y(x+y),
\end{equation}
 which  could be viewed as a special case of systems
\eqref{sys:Config.3.12-G}, when $f=0$. So considering
\eqref{expr:Delta-i,delta-i:3.12} the singular point $M_3$ has
coalesced   with $M_1(0,0)$ (becoming a saddle-node) and $M_4$ in
this case is a focus. Therefore considering Remark
\ref{rem:two-S-N}   we get univocally {Picture
3.13($\stackrel{\,*}{l}$1)}.

\subsection{The phase portraits of degenerate LV-systems}
\label{subs:Phase-Portraits-Degen-LV}

In this section we   examine   the phase portraits of the
degenerate LV-systems with the configurations {Configs.
LV$_d$.j} with $j=1,2,\ldots,14$ (see Table 4  and
Fig. \ref{Fig:Config-InfNs}).

\begin{theorem}\label{thm:Phase-Portraits-Degen-LV}
The degenerate LV-systems have a total of 20 topologically distinct phase
portraits which are given in Fig.\ref{Fig:Phase-Port-Deg}.  The necessary
and sufficient conditions for the realization of each one of these
phase portraits are given in columns 2 and 3 of Table 6.
\end{theorem}

\begin{table}[!htb]
\begin{center}
{\footnotesize
\begin{tabular}{|c|@{}c@{}|@{}c@{}|@{}c@{}|c|}
\multicolumn{5}{c}{Table 6}\\[1mm]
\hline $ \begin{gathered}{Configu-}\\
\emph{ration}\end{gathered} $ & $\begin{gathered}\emph{Necessary and suffi-}\\
\emph{cient conditions}
\end{gathered}$ & \multicolumn{2}{c|}{$ \begin{gathered}\emph{Additional conditions}\\ \emph{for phase
portraits}\end{gathered}$} & $\begin{gathered}\emph{Phase}\\ \emph{portrait}\end{gathered} $ \rule{0pt}{6.7mm}\\[0.4mm]
\hline
 \multirow{2}*{{Config. LV$_d$.1}} &   \multirow{2}*{$  \begin{gathered}\eta>0,
                  \mu_{0,1,2,3,4}=0,\\ \theta \ne0, H_7\ne0  \end{gathered} $}
    &  \multicolumn{2}{c|}{$\begin{gathered}  K<0\end{gathered}$} & {Picture LV$_d$.1(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-5}
    &   &\multicolumn{2}{c|}{$\begin{gathered}  K>0\end{gathered}$} & {Picture LV$_d$.1(b)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
 \multirow{2}*{{Config. LV$_d$.2}} &   \multirow{2}*{$  \begin{gathered}\eta>0,
                  \mu_{0,1,2,3,4}=0,\\ \theta \ne0, H_7=0  \end{gathered} $}
    &  \multicolumn{2}{c|}{$\begin{gathered}  K<0\end{gathered}$} & {Picture LV$_d$.2(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-5}
    &   &\multicolumn{2}{c|}{$\begin{gathered}  K>0\end{gathered}$} & {Picture LV$_d$.2(b)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
 \multirow{1}*{{Config. LV$_d$.3}} &    {$  \begin{gathered}\eta>0,
                  \mu_{0,1,2,3,4}=0,\\ \theta =H_4=0, H_7\ne0 \end{gathered} $}
    &  \multicolumn{2}{c|}{$-$} & {Picture LV$_d$.3}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
 \multirow{1}*{{Config. LV$_d$.4}} &    {$  \begin{gathered}\eta>0,
                  \mu_{0,1,2,3,4}=0,\\ \theta =H_4=0, H_7=0 \end{gathered} $}
    &  \multicolumn{2}{c|}{$-$} & {Picture LV$_d$.4}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
 \multirow{1}*{{Config. LV$_d$.5}} &    {$  \begin{gathered}\eta=0,
                  \mu_{0,1,2,3,4}=0,\\ \theta\ne0, H_7\ne0 \end{gathered} $}
    &  \multicolumn{2}{c|}{$-$} & {Picture LV$_d$.5}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
 \multirow{1}*{{Config. LV$_d$.6}} &    {$  \begin{gathered}\eta=0,
                  \mu_{0,1,2,3,4}=0,\\ \theta\ne0, H_7=0 \end{gathered} $}
    &  \multicolumn{2}{c|}{$-$} & {Picture LV$_d$.6}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
 \multirow{3}*{{Config. LV$_d$.7}} &   \multirow{3}*{$  \begin{gathered}\eta=0,
                  \mu_{0,1,2,3,4}=0,\\ \theta =0,K\ne0, H_2\ne0  \end{gathered} $}
    &  \multicolumn{2}{c|}{$\begin{gathered}  K<0\end{gathered}$} & {Picture LV$_d$.7(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-5}
    &   &\multirow{2}*{$\begin{gathered}  K>0\end{gathered}$}&$L<0$ & {Picture LV$_d$.7(b)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{4-5}
    &   &   &$L>0$ & {Picture LV$_d$.7(c)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
 \multirow{3}*{{Config. LV$_d$.8}} &   \multirow{3}*{$  \begin{gathered}\eta=0,
                  \mu_{0,1,2,3,4}=0,\\ \theta =0,K\ne0, H_2=0  \end{gathered} $}
    &  \multicolumn{2}{c|}{$\begin{gathered}  K<0\end{gathered}$} & {Picture LV$_d$.8(a)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{3-5}
    &   &\multirow{2}*{$\begin{gathered}  K>0\end{gathered}$}&$L<0$ & {Picture LV$_d$.8(b)}\rule{0pt}{3.8mm}\\[0.3mm]
 \cline{4-5}
    &   &   &$L>0$ & {Picture LV$_d$.8(c)}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
 \multirow{1}*{{Config. LV$_d$.9}} &    {$  \begin{gathered}\eta=0,
                  \mu_{0,1,2,3,4}=0,  \theta=0,\\ K=H_7=0,   N\ne0,  H_2\ne0 \end{gathered} $}
    &  \multicolumn{2}{c|}{$-$} & {Picture LV$_d$.9}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
 \multirow{1}*{{Config. LV$_d$.10}} &    {$  \begin{gathered}\eta=0,
                  \mu_{0,1,2,3,4}=0,  \theta=0,\\ K=H_7=0,   N\ne0,  H_2=0 \end{gathered} $}
    &  \multicolumn{2}{c|}{$-$} & {Picture LV$_d$.10}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
 \multirow{1}*{{Config. LV$_d$.11}} &    {$  \begin{gathered}\eta=0,
                  \mu_{0,1,2,3,4}=0,  \theta=0,\\ K= N= D=N_1=0,  N_5>0 \end{gathered} $}
    &  \multicolumn{2}{c|}{$-$} & {Picture LV$_d$.11}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
 \multirow{1}*{{Config. LV$_d$.12}} &    {$  \begin{gathered}\eta=0,
                  \mu_{0,1,2,3,4}=0,  \theta=0,\\ K= N= D=N_1=0,  N_5=0 \end{gathered} $}
    &  \multicolumn{2}{c|}{$-$} & {Picture LV$_d$.12}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
 \multirow{1}*{{Config. LV$_d$.13}} &    {$  \begin{gathered}C_2=0,
                  \mu_{0,1,2,3,4}=0,    H_2\ne0 \end{gathered} $}
    &  \multicolumn{2}{c|}{$-$} & {Picture LV$_d$.13}\rule{0pt}{3.8mm}\\[0.3mm]
\hline
 \multirow{1}*{{Config. LV$_d$.14}} &    {$  \begin{gathered}C_2=0,
                  \mu_{0,1,2,3,4}=0,    H_2=0 \end{gathered} $}
    &  \multicolumn{2}{c|}{$-$} & {Picture LV$_d$.14}\rule{0pt}{3.8mm}\\[0.3mm]
  \hline
\end{tabular}
}\end{center}
\end{table}


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.95\textwidth]{fig6} % Portraits-LV-deg-4
\end{center}
\caption{Phase portraits of the family of degenerate
LV-systems}\label{Fig:Phase-Port-Deg}
\end{figure}


\begin{proof}[Proof of Theorem \ref{thm:Phase-Portraits-Degen-LV}]
 We   examine each one of the canonical systems (LV$_d$.j)
($j\in\{1,2,\ldots,14$) given in the Table 6
corresponding to the configurations Config. LV$_d$.j of the
degenerate LV-systems.

Clearly a degenerate real LV-system  must posses  at least one
real affine straight line filled up with singularities. So the
phase portraits can easily be detected and in what follows we only
indicate for a given configuration: (i) the invariant lines filled
up with singularities; (ii) the corresponding linear (or even
constant) systems; (iii) the invariant lines of the linear
systems; (iv) the topologically distinct phase portraits of the
respective quadratic systems; (v)  and whenever necessary the
affine invariant polynomials which provide the respective
conditions.

\subsubsection{\normalsize The phase portraits associated with
{Config. LV$_d$.1} }

(i)   Singular line:   $x=0$;\\
(ii) Corresponding
linear systems:  $\dot x= 1+gx-y, \   \dot y= (g-1)y, \, g(g-1) \ne 0$;
\\
(iii) Invariant lines of the linear systems:
$y=0$ and $g(x-y)+1=0$;
\\
(iv) Phase portraits:  {Picture LV$_d$.1(a)} if $g(g-1)<0$ 
and {Picture LV$_d$.1(b)} if $g(g-1)>0$;
\\
(v) Invariant polynomial:
$K=2g(g-1)x^2$ $\Rightarrow$  $\operatorname{sign}(K) =\operatorname{sign}\big(g(g-1)\big)$.

\subsubsection{The phase portraits associated with {Config. LV$_d$.2} }

(i)  Singular line:   $x=0$;\\
(ii) Corresponding
linear systems:  $\dot x=  gx-y, \  \dot y= (g-1)y, \ g(g-1)\ne0$;
\\
(iii) Invariant lines of the linear systems:
$y=0$ and $ x-y =0$;
\\
(iv) Phase portraits:  {Picture LV$_d$.2(a)} if $g(g-1)<0$ and
{Picture LV$_d$.2(b)} if $g(g-1)>0$;
\\
(v) Invariant polynomial:
$K=2g(g-1)x^2$ $\Rightarrow$  $\operatorname{sign}(K) =\operatorname{sign}\big(g(g-1)\big)$.

\subsubsection{The phase portraits associated with {Config. LV$_d$.3}}
 (i)   Singular line:   $x=0$;\\
 (ii) Corresponding
linear system:\  $\dot x=  1+y, \ \ \dot y= y$;
\\
(iii) Invariant lines of the linear system:
$y=0$;
\\
(iv) Phase portrait:   {Picture LV$_d$.3}.

\subsubsection{The phase portraits associated with {Config. LV$_d$.4}}
 (i)   Singular lines:   $x=0$ and $y=0$;\\
 (ii) Corresponding constant system:  $\dot x=  1, \  \dot y= 1$;
\\
(iii) Invariant lines of the constant system: $y=x+C$;
\\
(iv) Phase portrait:  {Picture LV$_d$.4}.

\subsubsection{The phase portraits associated with {Config. LV$_d$.5} }
 (i)   Singular line:  $y=0$;\\
  (ii) Corresponding linear system:  $\dot x=  x, \  \dot y= 1-x+y$;
\\
(iii) Invariant lines of the linear system: $x=0$ (double);
\\
(iv) Phase portrait:   {Picture LV$_d$.5}.

\subsubsection{The phase portraits associated with {Config. LV$_d$.6}}
 (i)   Singular line:  $y=0$;
 (ii) Corresponding linear system:  $\dot x=  x, \  \dot y=  -x+y$;
\\
(iii) Invariant lines of the linear system: $x=0$ (double);
\\
(iv) Phase portrait: {Picture LV$_d$.6}.

\subsubsection{The phase portraits associated with {Config. LV$_d$.7} }
 (i)   Singular line:   $x=0$; \\
 (ii) Corresponding linear systems:  $\dot x=  1+gx, \  \dot y= (g-1)y, \
g(g-1)\ne0$;
\\
(iii) Invariant lines of the linear systems: $y=0$ and $ gx+1 =0$;
\\
(iv) Phase portraits:   {Pictures: LV$_d$.7(a)} if $g(g-1)<0$;
 LV$_d$.7(b) if $g<0$;   LV$_d$.7(c) if $g>1$;
\\
(v) Invariant polynomials:
  $\begin{cases} K=2g(g-1)x^2 & \Rightarrow 
 \operatorname{sign}(K)=\operatorname{sign}\big(g(g-1)\big);\\
L=8gx^2 & \Rightarrow   \operatorname{sign}(L) =\operatorname{sign} (g).
\end{cases}$

\subsubsection{The phase portraits associated with {Config. LV$_d$.8} }

 (i)   Singular line:   $x=0$;\\
 (ii) Corresponding linear systems:
$\dot x=  gx, \  \dot y= (g-1)y, \ g(g-1)\ne0$;
\\
 (iii) Invariant lines of the linear systems:\ $y=0$ and $ x =0$;
\\
(iv) Phase portraits:   {Pictures: LV$_d$.8(a)} if $g(g-1)<0$;
 LV$_d$.8(b) if $g<0$;  LV$_d$.8(c) if $g>1$;
\\
(v) Invariant polynomials:
$ \begin{cases} K=2g(g-1)x^2 & \Rightarrow 
\operatorname{sign}(K)=\operatorname{sign}\big(g(g-1)\big);\\
L=8gx^2 & \Rightarrow   \operatorname{sign}(L) =\operatorname{sign} (g).
\end{cases}$

\subsubsection{The phase portraits associated with {Config. LV$_d$.9} }

 (i)   Singular line:   $x=0$; \\
 (ii) Corresponding linear system:\  $\dot x=  1, \  \dot y=   y$;
\\
(iii) Invariant lines of the linear system: $y=0$;
\\
(iv) Phase portrait:  {Picture LV$_d$.9}.

\subsubsection{The phase portraits associated with {Config. LV$_d$.10} }
 (i)   Singular lines:   $x=0$ and $y=0$;\\
 (ii) Corresponding constant system:  $\dot x=  0, \  \dot y=   1$;
\\
(iii) Invariant lines of the constant system: $x=C$, $C\in\mathbb{R}$;
\\
(iv) Phase portrait:   {Picture LV$_d$.10}.

\subsubsection{The phase portraits associated with {Config. LV$_d$.11}}
 (i)   Singular lines:   $x=0$ and $x+2=0$; \\
 (ii) Corresponding constant system:  $\dot x=  1, \  \dot y=   0$;
\\
(iii) Invariant lines of the constant system: $y=C$, $C\in\mathbb{R}$;
\\
(iv) Phase portrait:  {Picture LV$_d$.11}.

\subsubsection{\normalsize The phase portraits associated with {Config. LV$_d$.12} }
 (i)   Singular line:   $x^2=0$;\\
 (ii) Corresponding constant system:  $\dot x=  1, \  \dot y=   0$;
\\
(iii) Invariant lines of the constant system: $y=C$, $C\in\mathbb{R}$;
\\
(iv) Phase portrait:   {Picture LV$_d$.12}.

\subsubsection{\normalsize The phase portraits associated with {Config. LV$_d$.13} }
 (i)   Singular line:   $x=0$; \\
 (ii) Respective linear system:  $\dot x=  1+x,  \ \dot y=   y$;
\\
(iii) Invariant lines of the linear
system: $y=C(x+1)$, $C\in\mathbb{R}$;
\\
(iv) Phase portrait:  {Picture LV$_d$.13}.

\subsubsection{\normalsize The phase portraits associated with {Config. LV$_d$.14} }
 (i)   Singular line:   $x=0$;\\
 (ii) Corresponding linear system:  $\dot x=  x, \  \dot y=   y$;
\\
 (iii) Invariant lines of the linear system:\ $y=Cx$, $C\in\mathbb{R}$;
\\
(iv) Phase portrait:  {Picture LV$_d$.14}.

\subsection{Topologically distinct phase portraits of LV-systems}\label{sec:topol-distinct}

To find the exact number of topologically distinct phase
portraits of LV-systems, we use a number of topological
invariants for  distinguishing (or identifying)  phase portraits.
We list below the topological invariants we need and the notation
we use.


\noindent I. \emph{Singularities, invariant lines, multiplicities   and indices:}
\begin{itemize}
    \item $\mathcal{N}$ = total number of all singularities (they are all real) of the
    systems;
    \item $\begin{pmatrix} \mathcal{N}_f\\ \mathcal{T}_m\end{pmatrix}$
 = the number $ \mathcal{N}_f$
    of all distinct  finite singularities having a total multiplicity $\mathcal{T}_m$;
    \item $\deg  J$ = the sum of the indices  of all finite singularities   of the
    systems;
    \item $\mathcal{N}^{\rm sing}_{\rm AIL}$ = total number of affine invariant lines filled up with
    singularities;
    \item $\mathcal{N}_{\infty}$ = total number of infinite singularities;
\end{itemize}

\noindent II. \emph{Connections of separatrices:}
\begin{itemize}
   \item $\#SC^s_s$ = total number of connections of a finite saddle to a finite saddle;
   \item $\#SC^S_s$ = total number of connections of a finite saddle to an infinite saddle;
   \item $\#SC^\mathcal{S N}_s$ =total number of connections of a finite saddle to an infinite saddle-node;
   \item $\#SC^s_{sn}$ = total number of connections of a finite  saddle-node  to a finite saddle;
   \item $\#SC^S_{sn}$ = total number of connections of a finite saddle-node  to an infinite saddle;
   \item $\#SC^\mathcal{S N}_{sn}$ = total number of connections of a finite saddle-node  to an infinite saddle-node;
   \item $\#SC^\mathcal{S}_{sn(hh)}$ = total number of separatrices  dividing
          the two hyperbolic sectors of finite saddle-nodes,   going to infinite saddles;
   \item $\#SC^\mathcal{S N}_{sn(hh)}$ = total number of separatrices dividing
          the two hyperbolic sectors of finite saddle-nodes connecting with separatrices of  infinite saddle-nodes.
   \item $\#Sep^\mathcal{\,S N}_{(H H)}$ = total number of separatrices of infinite saddle-nodes
          located in the finite plane and dividing the two hyperbolic sectors.
\end{itemize}


\noindent III. \emph{The   number of separatrices or orbits leaving from or ending
at a singular point:}
\begin{itemize}
 \item $M_{\rm sep}^{\tilde n}$ = $\max \{\operatorname{sep}(\tilde n)|\, \tilde n\ \text{is a node} \}$, where
         $\operatorname{sep}(\tilde n)$ is the number of separatrices leaving from or ending  at a finite node $\tilde n$;

\item $M_{\rm sep}^{\tilde s\tilde n}$ = $\max \{\operatorname{sep}(\tilde s\tilde n)|\, \tilde s\tilde n\ \text{is a node} \}$, where
         $\operatorname{sep}(\tilde s\tilde n)$ is the number of separatrices leaving from or ending at a finite saddle-node $\tilde s\tilde n$;

\item $M_{\rm orb}$ = $\max \{\operatorname{orb}(p)|\, p\ \text{is a finite singularity} \}$,
  where $\operatorname{orb}(p)$ is the number of orbits  leaving from or arriving at  $p$;

\item $M_{\rm ORB}$ = $\max \{\operatorname{orb}(p_1,p_2)|\, p_1,p_2\ \text{are infinite singularities}\}$,
  where $\operatorname{orb}(p_1,p_2)$ is the number of orbits connecting  $ p_1$ with
  $p_2$.
   \end{itemize}

Using the topological invariants listed above we construct the
following global topological invariant
$\mathcal{I}=(\mathcal{I}_1,\mathcal{I}_2, \mathcal{I}_3$), where
\begin{align*}
\mathcal{I}_1&=\Big(\mathcal{N},  
\begin{pmatrix} \mathcal{N}_f\\ \mathcal{T}_m\end{pmatrix},
 \deg  J,  \mathcal{N}^{\rm sing}_{I L A},
\mathcal{N}_{\infty} \Big),\\
\mathcal{I}_2&=\Big(\#SC^s_s, \#SC^S_s, \#SC^\mathcal{S N}_s, \#SC^s_{sn},
\#SC^S_{sn},
\#SC^\mathcal{S N}_{sn}, \#SC^\mathcal{S}_{sn(hh)}, \#Sep^\mathcal{\,S N}_{(H H)} \Big),\\
\mathcal{I}_3&=\Big(M_{\rm sep}^{\tilde n},  M_{\rm sep}^{\tilde s\tilde n}, M_{\rm orb},  M_{\rm ORB} \Big),\\
\end{align*}
which classifies all LV-systems.
\end{proof}

\begin{table}[!htb]
\begin{center}
\includegraphics[width=0.9\textwidth]{diagram-n7h}
\end{center}
\caption{Global Topological   Diagram: phase
portraits of LV-systems with $N=7$} \label{Fig:N=7}
\end{table}



\begin{theorem}\label{thm:top distinct}
I. The class of non-degenerate LV-systems have a total of 92
topologically distinct phase portraits. The different phase
portraits are contained in the \emph{Global Topological
Diagrams} (see Diagrams \ref{Fig:N=7}--\ref{Fig:N=3})
distinguished by the various
components of $\mathcal{I}$. In the middle of these diagrams there
appear a total of 152 phase portraits for the classes (i)-(iii) of
the Main Theorem, and topological equivalences are listed. On the
right side of these diagrams the distinct phase portraits are
numbered from (1) to (92). Moreover for each phase portrait we
indicate on its right side the corresponding phase portraits in
the paper \cite{CaoJia08}. More precisely we have the following
three cases:

(a) to the portrait (i) there corresponds only one portrait in
\cite{CaoJia08};

(b) to the portrait (i) there correspond several portraits claimed
to be distinct in \cite{CaoJia08}.

(c) to the portrait (i) with $i\in \{68,81,86,87 \}$ there is no
corresponding phase portrait in \cite{CaoJia08}. So  from the 92
phase portraits 4 portraits are  missing in \cite{CaoJia08}, due
to the different  use of the notion of quadratic  Lotka-Volterra
systems in \cite{CaoJia08} (see Observation \ref{obs:dif-LVs}).

II. The class of  degenerate LV-systems have a total of 20
topologically distinct phase portraits, distinguished by the
topological invariant $\mathcal{I}$. They are numbered from (93)
to (112) in the diagram appearing in Fig.\ref{Fig:N=inf}.
For each phase portrait we indicate on its right side the
corresponding phase portraits in the paper \cite{CaoJia08} and the
possibilities (a) and (b) above occur  also here. Moreover we have


(c') to the portrait (j) with $j\in \{95,102,103,106
\}$ there is no corresponding phase portrait in \cite{CaoJia08}.
So from the 20 phase portraits of degenerate LV-systems, 4
portraits are missing in \cite{CaoJia08} due to the remark in
point (c) above.
\end{theorem}

\begin{proof} The phase portraits appearing in the Diagrams
\ref{Fig:N=7}--\ref{Fig:N=inf}
for which the values of some of the components of the topological 
invariant $I$ are different, clearly
cannot be topologically equivalent. We thus
only need to show that whenever for two phase portraits the
corresponding values listed in these diagrams of the components of
this invariant $I$ coincide and the portraits are indicated as
being equivalent, then they are indeed equivalent.

We know that in quadratic systems, inside a limit cycle we have a
unique singularity which is a focus. Although a node and a focus
are not distinguished by the topological equivalence relation,
this distinction is important for the possible presence of limit
cycles. So we wanted to keep this distinction in our diagrams. We
observe that in the Diagrams \ref{Fig:N=7}--\ref{Fig:N=inf} we
have couples of topologically equivalent phase portraits, which
are however distinguished because in one we have a focus where in
the other portrait we have a node. Hence we do not need to prove
the equivalences in (3), (8), (10), (11), (16), (19), (29), (32),
(33), (38), (40), (54) and (62).

In the remaining cases whenever a similar repetition occurs we
only need to consider one of the two topologically equivalent
portraits, for example the one with a node.  We then confront it
with the remaining portraits listed as being topologically
equivalent, and show that they are indeed equivalent.

To prove this we make use of the concept of \emph{separatrix
configuration} defined by Markus in \cite{Markus}, called the
\emph{completed separatrix skeleton} in \cite{DLA}. Roughly
speaking this is the set of all separatrices together with one
orbit from each canonical region.  Theorem 1.43 in \cite{DLA}
(Markus-Neumann-Peixoto Theorem) says that two continuous flows on
the plane with only isolated singular points are topologically
equivalent if and only if their completed separatrix skeletons
$CSS_1$ and $CSS_2$ are equivalent, i.e. there exists a
homeomorphism of the plane mapping the orbits of $CSS_1$ to the
orbits of $CSS_2$. Furthermore according to \cite{Markus} instead
of having to prove the existence of a homeomorphism of $\mathbb{R}^2$
carrying the orbits of $CSS_1$ to the orbits of $CSS_2$, it
suffices to check that there is an isomorphism of the two
\emph{chordal systems} (see \cite{Kaplan}), which are the two
completed separatrix skeletons.


To prove equivalence of two portraits, we look at their
separatrices and canonical regions. After checking that we have
the same number of canonical regions we match them one by one and
we check that their bordering separatrices correspond. In some
cases, the equivalence is obvious, for example in case (1) where
the two portraits, which appear in
{Fig}.\,\ref{Fig:Ph-Port:4-6IL} and
{Fig}.\,\ref{Fig:Ph-Port-3ILs} are identical. We only need
to consider the remaining cases. We consider below a case for
which we prove the equivalence.  The other cases have been treated
in an entirely analogous way.

\begin{table}[!htb]
\begin{center}
\includegraphics[width=0.80 \textwidth]{diagram-n6c}
\end{center}
\caption{Global Topological Diagram: phase
portraits  of LV-systems with $N=6$} \label{Fig:N=6}
\end{table}


\begin{table}[!htb]
\begin{center}
\includegraphics[width=1.0\textwidth]{diagram-n5e}
\end{center}
\caption{Global Topological Diagram: phase
portraits  of LV-systems with $N=5$ }\label{Fig:N=5}
\end{table}


 \begin{table}[!htb]
\begin{center}
\includegraphics[width=0.85\textwidth]{diagram-n4b}
\end{center}
\caption{Global Topological Diagram:  phase
portraits  of LV-systems with $N=4$} \label{Fig:N=4}
\end{table}


 \begin{table}[!htb]
\begin{center}
\includegraphics[width=0.85\textwidth]{diagram-n3b}
\end{center}
\caption{Global Topological Diagram:  phase
portraits  of LV-systems with $N=3$} \label{Fig:N=3}
\end{table}



 \begin{table}[!htb]
\begin{center}
\includegraphics[width=0.85\textwidth]{diagram-ninftyf}
\end{center}
\caption{Global Topological Diagram: phase
portraits  of LV-systems with $N=\infty$} \label{Fig:N=inf}
\end{table}

Case (5), portraits 3.1(c3) and 4.1(a).  In both portraits we have
7 canonical regions. We start by matching the two canonical
regions $CR_1$ and $CR_2$ in the two portraits determined by the
separatrices of the finite saddles. These are the only canonical
regions which together with their limiting separatrices are
bounded in the plane and these regions have equivalent orbit
representatives. These two canonical regions are each bordered by
4 separatrices. We next look at the four canonical regions which
have a common border separatrix with $CR_1$, respectively $CR_2$
and check that they are of the same kind in both portraits which
indeed occurs. Finally we look at the remaining two canonical
regions whose borders have only one common point (a saddle) with
the borders of $CR_1$, respectively $CR_2$. Each one of these two
regions in $CR_1$ has a corresponding region in $CR_2$ and these
two regions have equivalent orbit representatives. Similar
arguments work for the numerous other equivalences listed in the
diagrams.

To prove the point (b) in Theorem 3.3 we consider here two cases:

(1) The phase portraits (14-1) and (25-3) from Fig.1 of
\cite{CaoJia08}. These portraits are claimed to be non-equivalent.
Indeed these two portraits occur in Fig.1 which has the caption:
"The 79 non-topologically equivalent phase portraits of vector
field $X_1$." These two phase portraits are however equivalent to
the {Picture  3.1(c3)}  (and to the {Picture 4.1(a)}
4.1(a)) of case   (5) discussed above. To convince ourselves of
this we repeat the process above starting by counting the number
of canonical regions which is again 7 in both cases (14-1) and
(25-3). We also have, in (14-1) (respectively in (25-3)) only one
canonical region which together with its boundary separatrices is
bounded in the plane and the respective orbit representatives have
the same behavior. Continuing the process described above for the
case (5) of Diagram  \ref{Fig:N=7} in this article, we see
that (14-1) and (25-3) in   Fig.1 of \cite{DLA} are topologically
equivalent and that they are both equivalent with {Picture
3.1(c3)} (or with {Picture 4.1(a)}) of Diagram
\ref{Fig:N=7} in this paper.

(2) In \cite{CaoJia08} the authors claimed that in Fig.1  the phase
portraits (31-1-1) and (31-1-2) are topologically  distinct. More
exactly in Remark 6.1 on the page  818 it is mentioned: "the only
difference of pictures (31-1-1) and (31-1-2) is the stability of
point  $P$" (which is a focus). As it follows from \cite{CaoJia08}
these pictures correspond to systems which  belong  to the family
$X_1$:
\begin{equation}\label{sys:X1}
\dot x=x(1+x+by),\quad \dot y  = y (c+dx+y), \quad b,c,d\in\mathbb{R},
\end{equation} 
when some restrictions on the parameters $b,c$ and $d$ are
imposed.

We claim that the phase portraits (31-1-1) and (31-1-2) in Fig.1
of  \cite{CaoJia08} are topologically equivalent. To prove this
we consider the following two steps:

(a) we take a specific system from the family \eqref{sys:X1}
corresponding to a  point $(b_0,c_0,d_0)$ fixed in the parameter
space, which  possesses the phase portrait (31-1-1);  and $b)$ we
construct a respective rescaling of the variables and time which
leads to a system with the phase portrait (31-1-2).

Thus we fix $(b_0,c_0,d_0)=(4,-2,3)$ and we consider the system
\begin{equation}\label{sys:X1-a}
\dot x=x(1+x+4y),\quad \dot y  = y (-2+3x+y).
\end{equation}
It is easy to detect (for example, using the program P4
(see \cite{DLA}) that  the phase portrait of this system is
exactly (31-1-1). Now applying the rescaling
$(x,y,t)\mapsto(-y,-x,-t)$ we get the system
\begin{equation}\label{sys:X1-b}
\dot x=x(2+x+3y),\quad \dot y  = y (-1+4x+y),
\end{equation}
the phase portrait of which correspond to (31-1-2).

We note that a similar rescaling could be applied for the whole
family \eqref{sys:X1} in the case of picture (31-1-1) and this
leads to the systems with the phase portrait (31-1-2). Thus our
claim is proved.

Similar arguments as those encountered in 1) and 2) above hold for
all the remaining cases.

We point out that in \cite{CaoJia08} the authors work with the
restriction that both polynomials $p(x,y)$ and $q(x,y)$ in
\eqref{sys:gen-LV} are of degree 2. For this reason, in
\cite{CaoJia08} there are some missing portraits which we have
here (recall that we ask here only for at least one of the
polynomials $p$ and $q$ to be of degree 2). It can be easily
checked that the portraits indicated in the points $c$ and $c'$ in
the theorem are indeed missing in \cite{CaoJia08}. For the first
part of II of the theorem, the proof is easy as the systems are
degenerate and once we remove the common factor of $p(x,y)$ and
$q(x,y)$ we have systems which are linear or they have constant
right sides.
\end{proof}

\begin{remark} \rm
We observe that in the \emph{Global Topological Diagrams} on the
extreme right hand side we have occasionally a star.  For example
in the diagram from Diagram \ref{Fig:N=7} for the portrait
(9) we have [Fig.1: (24-3$)^*$]. We use the star to
indicate those cases where some mistake occurs in that phase
portrait, such as for example a wrong orientation of a specific
phase curve, or the presence of a phase curve which should not be
there or the absence of some separatrices or some other minor
error. If such a mistake is corrected, then the resulting phase
portrait is equivalent to the corresponding phase portrait in the
middle of the diagram.
\end{remark}


\subsection{Concluding remarks}
We sum up in the next theorem some basic geometric global
properties of the class of LV-systems.

\begin{theorem}\label{thm:conclud-comments}
Consider an LV-system $(S)$. I. Then $(S)$
\begin{itemize}
\item[(1)] has only real invariant lines;
\item[(2)] has only real singularities, at least two of them at infinity;
\item[(3)] has no finite singularities of multiplicity three;
\item[(4)] has a focus only if $(S)$ has exactly three invariant lines, all simple;
\item[(5)] has no weak foci;
\item[(6)] has no limit cycles.
\end{itemize}
II. In the generic case when the system  $(S)$ has exactly three
invariant lines  all simple,  $(S)$ has no centers.
\end{theorem}

\begin{proof}  \emph{I}. The points (1) and (2) easily
follow from the definition of LV-systems and from the normal form
\eqref{sys:gen-LV}. The point   (3) was proved in
\cite[p. 187]{SchVul10-JFPTA}  (also in \cite{SchNai08}).

Point  (4). All LV-systems with invariant lines of
total multiplicity at least four do not have a focus (see
{Fig}. \ref{Fig:Ph-Port:4-6IL}). Similarly the phase
portraits with all points at infinity singular (see Fig.
\ref{Fig:Pictures:C2-0}) as well as the degenerate LV-systems
(see {Fig}. \ref{Fig:Phase-Port-Deg}) have no foci.


 Points (5) and (6) were proved in Lemma \ref{lem:M4-center} and
Theorem \ref{thm:Bautin} respectively.

\emph{II}. From the Table 2 it follows that for  an LV-system
with exactly three invariant lines all simple, the condition
$B_3\ne0$ holds. On the other hand by Lemma \ref{lem:M4-center}
for the existence of a center the condition $B_3=0$ is necessary
and this contradiction completes the proof of the statement.
\end{proof}

\begin{theorem}\label{thm:graphics}
I. Of 112 topologically distinct phase portraits of LV-systems
only 18 possess graphics and all of them occur  in
${\rm QSL}_{ i}$, $i\in\{3,4\}$. More precisely we have:
\begin{itemize}
\item[(i)] 8 distinct isolated graphics occur in systems with exactly three  invariant lines,
 all   simple. All of them are triangles   with an infinite side and they
 surround a focus.

\item[(ii)] 4 distinct isolated graphics occur in systems  in  ${\rm QSL}_{4}$  
 all of them are triangles, one finite and three with an
infinite side and they surround a center.

\item[(iii)] non-isolated    graphics occur in 6 topological distinct phase 
portraits of  systems  in  ${\rm QSL}_{4}$. In each one of them we have two 
infinite families  of graphics. These graphics are: (a) homoclinic loops with 
either a finite singularity or with  an infinite singularity;
(b) limiting  triangles of families of  homoclinic loops.
\end{itemize}

II.  Infinite families of degenerate graphics occur in: (a)
LV-systems with all points at infinity singular, excepting the
systems with the phase portrait Picture $C_2.$5(a), and
(b) degenerate LV-systems.
\end{theorem}

\begin{proof}
 \emph{I}. The proof of the points (i) and
(ii) results from Fig. \ref{Fig:Ph-Port-3ILs}
and Fig. \ref{Fig:Ph-Port:4-6IL} respectively.

(iii) The proof results from
Fig. \ref{Fig:Ph-Port:4-6IL}. More precisely the only
phase portraits in this figure, which possess non-isolated
graphics are {Pictures 4.5(c), 4.20(b)} (these have
homoclinic loops with a finite singularity), 
{Pictures 4.12(b), 4.12(c), 4.19(a)} and 4.24(b) (these have
homoclinic loops with an infinite singular point).

\emph{II}. The proof of this part results from
Fig. \ref{Fig:Pictures:C2-0} and
Fig. \ref{Fig:Phase-Port-Deg}.
\end{proof}



\begin{thebibliography}{99}

\bibitem{Art-Llib}  {J.~C. Art\'es, J. Llibre};
\emph{Quadratic {H}amiltonian vector fields}, J. Differential
Equations \textbf{107} (1994), 80--95.

\bibitem{Art-Llib-Schl}  {J.~C. Art\'es, J. Llibre, D. Schlomiuk};
\emph{The geometry of quadratic dufferential systems with a weak
focus of second order}, International J. of Bifurcation and Chaos
\textbf{16} (2006), 3127-–3194.


\bibitem{ArtLliVul08-IJBCh} J.~C. Art\'es, J. Llibre, N. Vulpe;
\emph{Singular points of quadratic systems: a complete
classification in the coefficient space $\mathbb{R}^{12}$},
 International  Journal of Bifurcation Theory and Chaos,
Vol. \textbf{18}, No.2 (2008) 313--362.

\bibitem{ArtLliVul-Prep2011} J.~C. Art\'es, J. Llibre,  N. Vulpe;
 \emph{Quadratic systems with an integrable saddle:  A
complete classification in the coefficient space
$\mathbb{R}^{12}$},  Preprint, n\'um. 14, 2011,
 Universitat Aut\`onoma de Barcelona, 37 pp.

\bibitem{ArtLliVul-EJDE} J.~C. Art\'es, J. Llibre, N. Vulpe;
 \emph{Complete geometric invariant study of two classes of
quadratic systems},  Electronic J. Differential Equations,
Vol. 2012 (2012) No. 09, 1--35.

\bibitem{BalVul97-DEDS} V. A. Baltag, N. I.  Vulpe;
\emph{Total multiplicity of all finite critical points of the
polynomial differential system},  Planar nonlinear
dynamical systems (Delft, 1995). Differential Equations \& Dynam.
Systems $\mathbf 5$~(1997), no. 3-4, 455--471.


\bibitem{BalVul98-BASM} V. Baltag, N.  Vulpe;
\emph{The differential operators and
multiplicity of singular points
  for polynomial differential system}, Bul. Acad. \c Stiin\c te Repub. Mold.
  Mat. (1998), no.~1, 81--94, 151, 154.

\bibitem{Bautin} N. N. Bautin;
 \emph{On periodic solutions of a
system of differential equations} (R), Prikl. Mat. Mech.
\textbf{18} (1954), 128.

\bibitem{Berl} A. N. Berlinski\u i;
\emph{On the behaviour of integral curves of
a certain differential equation}, Izv. Vyss. Ucheb. Zaved. Mat.
1960, 2(15), pp. 3--18 (Russian). Translated into English by
National Lending Library for Science and Technology, Russian
Translating Programme RTS 5158, June 1969. Boston Spa, Yorkshire,
U.K.

\bibitem{BulCalTimVul96} D. Boularas, Iu. Calin, L. Timochouk, N. Vulpe;
\emph{T--comitants of
  quadratic systems: A study via the translation invariants}, Delft University
  of Technology, Faculty of Technical Mathematics and Informatics, Report no.
  96-90 (1996), (ftp://ftp.its.tudelft.nl/publications/tech-reports/1996/DUT-TWI- 96-90.ps.gz).


\bibitem{Chicone-Tian} C. Chicone, J. H. Tian;
 \emph{On general properties on quadratic systems}, Amr. Math. Monthly,
  \textbf{89} (1982), No. 3, 167-178.


\bibitem{ChrLliPer2007} C. Christopher, J. Llibre, V. Pereira;
 \emph{Multiplicity of invariant algebraic curves in polynomial
vector fields}, Pacific J. of Math. Vol. 229, No. 1, January 2007,
63-117.

\bibitem{CaoJia08} F. Cao,  J. Jiang;
\emph{The Classification on the Global Phase Portraits of
two-dimensional Lotka-Volterra system}, Dyn. Diff. Equat. (2008)
20, 797-830.

\bibitem{Cop} W. A. Coppel;
 \emph{A survey of Quadratic Systems},
J. Differential Equations, \textbf{2} (1966), 293--304.

\bibitem{Dar1878} G. Darboux;
\emph{M\'emoire sur les \'equations
    diff\'erentielles du premier ordre et du premier
    degr\'e}, Bulletin de Sciences Math\'ematiques, \textbf{2me s\'erie},
  2 (1) (1878), 60-96; 123-144; 151-200.

\bibitem{Dum-Fid}  F. Dumortier, P. Fiddelaers;
\emph{Quadratic models for generic local 3-parameter bifurcations
on the plane}, Trans. Am. Math. Soc. 326:101-126, 1991.

\bibitem{DLA} F. Dumortier, J. Llibre, J. C. Art\'es;
\emph{Qualitative Theory of Planar Differential Systems}, Springer, 2006, 298pp.


\bibitem{Geo07}  R.-M. Georgescu;
\emph{Group theory applications to the bifurcation study for some dynamical models},
Ph.D. thesis, University of Pitesti, Romania (2007) (in Romanian).

\bibitem{GraYou41}
Grace, J.~H., Young~A.;  \emph{The algebra of invariants}, New
York: Stechert,   1941.

\bibitem{Kaplan} Kaplan, W.; \emph{Regular curves families filling
the plane, I and II}, Duke Math. J. vol.7 (1940), pp. 154-185;
vol.8 (1941) pp. 11-46.

\bibitem{Llibre-Schlomiuk:2004}  J. Llibre, D. Schlomiuk;
\emph{The geometry of quadratic differential systems with a weak
focus of third order}, Canadian J. Math.  \textbf{56} (2004),
{310--343}.

\bibitem{Lot25} A. J.  Lotka;
 \emph{Elements of Physical Biology}, Waverly Press
by the Williams and Wilkins Company, Baltimore, Md., U. S. A.,
1925.

\bibitem{Markus} I. Markus;
 \emph{Global structure of ordinary differential
equations in the plane}.  Trans. AMS. \textbf{76} (1954), 127-148.


\bibitem{NikVul}  {I. Nikolaev, N. Vulpe};
\emph{Topological classification of quadratic systems at
infinity}, Journal of the London Mathematical Society, $\mathbf
2$~(1997), no. 55, 473-488.


\bibitem{Olv99} {P. J. Olver};
\emph{ Classical Invariant Theory},  London Mathematical Society
student texts: \textbf{ 44}, Cambridge University Press, 1999.

\bibitem{Pal-Schlomiuk:1997} {J. Pal, D. Schlomiuk};
\emph{Summing up the dynamics of  quadratic Hamiltonian systems
with a center}, Canadian J. Math.  \textbf{49} (1997), {583--599}.

\bibitem{Rey87} J. W. Reyn;
 \emph{Phase portraits of a quadratic system of differential equations occuring frequently in
applications}, Nieuw Archief voor Wiskunde, 5(1987), 107-151.

\bibitem{Roussarie} R.  Roussarie;
\emph{Smoothness property  for bifurcation diagrams}, Proceedings
of the Symposium on Planar Vector Fields (Lleida, 1996). Publ.
Mat. {\bf41} (1997), no. 1, 243-268.

\bibitem{Schlomiuk:1993} D. Schlomiuk;
\emph{Algebraic particular integrals, integrability and the
problem of the center}, {Trans. Amer. Math. Soc.} \textbf{338}
(1993), {799--841}


\bibitem{SchNai08} D. Schlomiuk, E. Naidenova;
 \emph{On the classification of Lotka-Volterra systems},
CRM Montreal, CRM Report  CRM-3266, September 2008, 15 pp.

\bibitem{SchVul04-QTDS}  D. Schlomiuk,  N. Vulpe;
\emph{Planar quadratic differential systems with invariant
straight lines of at least five total multiplicity}, Qualitative
Theory of Dynamical Systems, $\mathbf 5$~(2004), 135--194.

\bibitem{SchVul05-JDE}  D. Schlomiuk,  N. Vulpe;
\emph{Geometry of quadratic differential systems in the
neighborhood of infinity},  J. Differential Equations, \textbf{215}
(2005), 357--400.


\bibitem{SchVul08-RMJM}  D. Schlomiuk,  N. Vulpe;
\emph{Integrals and phase portraits of planar quadratic
differential systems with invariant lines  of at least five  total
multiplicity}, Rocky Mountain Journal of Mathematics, Vol.
38(2008), No. 6, 1--60


\bibitem{SchVul08-NATMA}  D. Schlomiuk,  N. Vulpe;
\emph{Planar quadratic differential systems with invariant
straight lines of  total multiplicity four}, Nonlinear Anal.,
2008, \textbf{68}, No. 4,  681--715


\bibitem{SchVul08-BASM}  D. Schlomiuk,  N. Vulpe;
 \emph{Integrals
and phase portraits of planar quadratic differential systems with
invariant lines  of total multiplicity four},  Bull. of Acad. of
Sci. of Moldova. Mathematics, No. 1(56), 2008, 27--83.

\bibitem{SchVul08-JDDE}  D. Schlomiuk,  N. Vulpe;
 \emph{The full
study of planar  quadratic differential systems possessing  a line
of singularities at infinity}, Journal of Dynamics and Diff.
Equations, Vol. 20(2008), No.4, 737--775.


\bibitem{SchVul10-JFPTA}  D. Schlomiuk,  N. Vulpe;
\emph{Global classification of the planar Lotka--Volterra
differential systems according to their configurations of
invariant straight lines }, Journal of Fixed Point Theory and
Applications, $\mathbf 8$~(2010), No.1, 69 pp.


\bibitem{Sib88-Mon}  K. S. Sibirskii;
\emph{Introduction to the algebraic theory of invariants of
differential equations}, Translated from the Russian. Nonlinear
Science: Theory and Applications. Manchester University Press,
Manchester, 1988.

\bibitem{Vol26} Vito Volterra;
 \emph{Variazione flutuazioni del numero d'individus in specie
animali conventi}, Mem. R. Acad. Naz. dei Lincei, 2(1926), 31-113.

\bibitem{Vulpe-DU} N. I. Vulpe;
\emph{Affine--invariant conditions for the topological
discrimination of quadratic systems with a center}, Differential
Equations \textbf{19} (1983), {273--280}.


\bibitem{Vul86-Book} N. I. Vulpe;
\emph{ Polynomial bases of comitants of differential systems and
their applications in qualitative theory}, (Russian)
``Shtiintsa'', Kishinev, 1986.


\bibitem{Wor-Bus93} W\"orz-Buserkros, A.;
 \emph{A complete classification of all
two-dimensional Lotka-Volterra Systems}, Diff.Eq. and Dyn.Syst.,
1(1993), 101-119

\bibitem{Zoladek:1994} {H. \.{Z}o\l\c{a}dek};
\emph{Quadratic systems with center and their perturbations}, {J.
Differential Equations}  \textbf{109} (1994), {223--273}.

\end{thebibliography}


\end{document}