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\AtBeginDocument{{\noindent\small 
\emph{Electronic Journal of Differential Equations}, 
Vol. 2012 (2012), No. 09, pp. 1--35.\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-2012/09\hfil Complete geometric invariant]
{Complete geometric invariant study of two classes of quadratic systems}

\author[J. C. Art\'es, J. Llibre, N. Vulpe \hfil EJDE-2012/09\hfilneg]
{Joan C. Art\'es, Jaume Llibre, Nicolae Vulpe}  % in alphabetical order

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

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

\address{Nicolae Vulpe \newline
Institute of Mathematics and Computer Science,
Academy of Science of Moldova, 
5 Academiei str, Chi\c sin\u au, MD-2028, Moldova}
\email{nvulpe@gmail.com}

\thanks{Submitted May 16, 2011. Published January 13, 2012.}
\subjclass[2000]{34C05, 34A34}
\keywords{Quadratic vector fields; weak singularities; type of singularity}

\begin{abstract}
 In this article, using affine invariant conditions,
 we give a complete study for quadratic systems with center
 and for quadratic Hamiltonian systems.
 There are two improvements over the results in \cite{Vulpe-DU} that
 studied centers up to $GL$-invariant, and over the results in
 \cite{Art-Llib} that classified Hamiltonian quadratic systems
 without invariants.
 The geometrical affine invariant study presented here is a crucial
 step toward the goal of the invariant classification of all quadratic
 systems according to their singularities, finite and infinite.
\end{abstract}

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

\newcommand{\is}{\mbox{\scriptsize{\sl \$}}}
\newcommand{\ses}{\raisebox{0.2em}[0pt][0pt]{${}_\emptyset$}}



\section{Introduction and statement of results}

Let $\mathbb{R}[x, y]$ be the ring of the polynomials in the variables $x$
and $y$ with coefficients in $\mathbb{R}$. We consider a system of
polynomial differential equations, or simply a polynomial
differential system, in $\mathbb{R}^2$ defined by
\begin{equation}\label{sis1}
\begin{gathered}
\dot{x}=P(x,y),\\
\dot{y}=Q(x,y),
\end{gathered}
\end{equation}
where $P,Q \in \mathbb{R}[x, y]$.
We say that the maximum of the degrees of
the polynomials $P$ and $Q$ is the degree of system
\eqref{sis1}. A \emph{quadratic polynomial differential system} or
simply a \emph{quadratic system} (QS) is a polynomial differential
system of degree $2$. We say that the quadratic system \eqref{sis1}
is \emph{non-degenerate} if the polynomials $P$ and $Q$ are
relatively prime or coprime; i.e., g.c.d. $(P,Q)= 1$.

During the previous one-hundred years quadratic vector fields have
been investigated intensively as one of the easiest but far from
trivial families of nonlinear differential systems, and more than
one thousand papers have been published about these vectors fields
(see for instance \cite{Reyn, Ye2, Ye}). However, the problem of
classifying all the  quadratic vector fields (even integrable
ones) remains open. For more information on the integrable
differential vector fields in dimension $2$, see for instance
\cite{Chava,Ll}.

Poincar\'e \cite{Poincare:1891} defined the notion of a
\emph{center} for a real polynomial differential system in the
plane (i.e. an isolated singularity surrounded by periodic
orbits). The analysis of the limit cycles which bifurcate from a
focus or a center of a quadratic system was made by Bautin
\cite{Bautin:1962}, by providing the structure of the power series
development of the displacement function defined near a focus or a
center of a quadratic system. More recently
the structure of this displacement function has
been understood for any weak focus of a polynomial differential
system. More precisely, first by using a linear change of
coordinates and a rescaling of the independent variable, we
transform any polynomial differential system having a weak focus
or a center at the origin with eigenvalues $\pm a i\neq 0$ (i.e.
having a \emph{weak focus}) into the form
\begin{equation}\label{ja}
\begin{gathered}
\dot x= y + P(x,y),\\
\dot y= -x + Q(x,y),
\end{gathered}
\end{equation}
where $P$ and $Q$ are polynomials without constant and linear terms.
Then the \emph{return map} $x\mapsto h(x)$ is defined for $|x|< R$,
where $R$ is a positive number sufficiently small to insure that the
power series expansion of $h(x)$ at the origin is convergent. Of
course, limit cycles correspond to isolated zeros of the
\emph{displacement function} $d(x)= h(x)-x$. The structure of the
power series for the displacement function is given by the following
restatement of Bautin's fundamental result (see \cite{Ro} for more
details): There exists a positive integer $m$ and a real number
$R>0$ such that the displacement function in a neighborhood of the
origin for the polynomial differential system \eqref{ja} can be
written as
\[
d(x)= \sum_{j=1}^m v_{2j+1} x^{2j+1} \Big[
\alpha_0+\sum_{k=1}^{\infty} \alpha_k^{2j+1} x^k\Big],
\]
for $|x|< R$, where the $v_{2j+1}$'s and the $\alpha_k^{2j+1}$'s
are homogeneous polynomials in the coefficients of the polynomials
$P$ and $Q$.

The constants $V_j=v_{2j+1}$ are called the \emph{focus
quantities} or the Poincar\'e-Liapunov constants. A weak focus for
which $V_1= \dots =V_{n-1}=0$ and $V_{n}\neq 0$ is a \emph{weak
focus of order $n$}. If all the focus quantities are zero then the
weak focus is a center. Note that any weak focus has finitely many
focus quantities, in our notation exactly $m$.

It is known that a polynomial differential system \eqref{ja} has a
center at the origin if and only if there exists a local analytic
first integral of the form $H=x^2+y^2+ F(x,y)$ defined in a
neighborhood of the origin, where $F$ starts with terms of order
higher than $2$. This result is due to Poincar\'e
\cite{Poincare:1891} (Moussu \cite{Mos} gave a geometrical proof
of this result).   Liapunov \cite{Li} extended   Poincar\'e
result  for the analytic case.

Through the coefficients of a quadratic system every one of these
systems can be identified with a single point of $\mathbb{R}^{12}$. One of
the first steps in a systematic study of the subclasses of $QS$ was
achieved in the determining the subclass $QC$ of all $QS$ having a
center. Of course this problem is algebraically solvable in the sense
indicated by Coppel \cite{Coppel:1966}, because the classification of
the quadratic centers is algebraically solvable.

The phase portraits of the class $QC$ were given by  Vulpe in
\cite{Vulpe-DU} and are here denoted by \emph{Vul}$_{\,\#}$ using his
classification. In that classification, only GL-invariants were used
which implied that systems could only be characterized after
displacing one center to the origin and adopting the standard normal
form. Later papers related with centers provide the bifurcation
diagrams for the different types of centers (see
\cite{Schlomiuk:1993,Zoladek:1994,Pal-Schlomiuk:1997}).

The polynomial differential system \eqref{sis1} is
\emph{Hamiltonian} if there exists a polynomial $H= H(x,y)$ such
that $P= \partial H/\partial y$ and $Q= -\partial H/\partial x$.
Regarding Hamiltonian systems, apart from many papers using them in
conservative systems, the first complete classification for
quadratic systems was done in \cite{Art-Llib} where quadratic
Hamiltonian systems were split into four normal forms and a
bifurcation diagram was provided for each one of them. No invariants
were used there. Later on in \cite{KalVul} the affine--invariant
conditions were established but they were constructed using
invariant polynomials of high  degree without explicit geometrical
meaning. In the later years the technique of the construction of
invariant polynomials has been greatly improved. Now, with these
better tools, the invariants needed are of lower degree and
consistent with the set of all invariants needed to describe
singular points.

The main results of this article are the following two theorems.

\begin{theorem}\label{thm:QS-center}
Consider a quadratic system of differential equations.
\begin{itemize}
\item[(i)] This system possesses a center and the
configuration of all its singularities (finite and infinite) up to
a congruent equivalence, given in Table \ref{table2} if and only if the
corresponding affine invariant conditions described in Table \ref{table2}
hold.

\item[(ii)] We have a total of 41 congruently distinct configurations of
singularities. For each phase portrait of quadratic systems with a
center we have the following two possibilities:
\begin{itemize}
\item[(a)]
 it corresponds to a unique configuration of
singularities; there are 17 such phase portraits;

\item[(b)] it corresponds to several configurations of
singularities; there are 14 such phase portraits. The richest
example is $Vul_2$  which could occur with anyone of the 8
congruently distinct configurations 15, 17, 19, 20, 26, 37, 39, 41.
\end{itemize}

\item[(iii)] The phase portrait of a  system with a center corresponds
to the one of 31 topologically distinct phase portraits constructed in
\cite{Vulpe-DU} (except the case of a linear system) and either it
is determined univocally by the respective configuration, or it is
determined by the configuration and additional conditions  given
in Table \ref{table3}. More exactly we have 35 congruently distinct
configurations, each of which  leads univocally to a unique phase
portrait; and there are 6 configurations each of which leads to
several phase portraits distinguished by the additional conditions
according to  Table \ref{table3}.
\end{itemize}
\end{theorem}

Tables \ref{table2} and \ref{table3} can be found in section
\ref{sec:centers}.

\begin{theorem}\label{thm:QS-Hamilt}
Assume that a quadratic system of
differential equations is Hamiltonian.
\begin{itemize}
\item[(i)] This system possesses the configuration of all its singularities
(finite and infinite) up to a congruent equivalence,   given in
Table \ref{table4} if and only if the corresponding affine invariant
conditions described in Table \ref{table4}  hold.

\item[(ii)] We have a total of 30 congruently distinct configurations of
singularities. For each phase portrait of Hamiltonian quadratic
systems  we have the following two possibilities:
\begin{itemize}
\item[(a)]
 it corresponds to a unique configuration of
singularities; there are 23 such phase portraits;

\item[(b)] it corresponds to several configurations of
singularities; there are 5 such phase portraits. The richest
example is $Ham_{11}$  which could occur with anyone of the 5
congruently distinct configurations 7,11, 20,26,28.
\end{itemize}

\item[(iii)] The phase portrait of this system   corresponds to the one
of 28 topologically distinct phase portraits constructed in
\cite{Art-Llib} and either it is determined univocally by the
respective configuration, or it is determined by the configuration
and additional conditions  given in Table \ref{table5}. More exactly we have
24 congruently distinct configurations, each of which  leads
univocally to a unique phase portrait; and there are 6
configurations each of which leads to several phase portraits
distinguished by the additional conditions according to
Table \ref{table5}.
\end{itemize}
\end{theorem}

Tables \ref{table4} and \ref{table5} can be found in
section \ref{sec:hamil}.

The work is organized as follows. In sections \ref{sec:equivalence}
and \ref{sec:Notation} we introduce the notation that we use for
describing the singular points. In section \ref{sec:preliminary} we
give some preliminary results needed for the work. In section
\ref{sec:diagram} we adapt a diagram from a previous paper
\cite{Blt-Vlp-BASM} to describe more easily the bifurcation tree of
finite singularities, and we also introduce the used invariants from
\cite{Vulpe-DU} and \cite{KalVul}. Finally in sections
\ref{sec:centers} and \ref{sec:hamil} we prove the main theorems of
this paper.


\section{Equivalence relations for singularities of planar
polynomial vector fields} \label{sec:equivalence}

We first recall the  topological equivalence relation as it is
used in most of the literature. Two singularities $p_1$ and $p_2$
are topologically equivalent if there exist open neighborhoods
$N_1$ and $N_2$ of these points and a homeomorphism $\Psi:
N_1\rightarrow {N_2}$ carrying orbits to orbits and preserving
orientation.  To reduce the number of cases, by topological
equivalence we shall mean here that the homeomorphism $\Psi$
\emph{preserves or reverses} the orientation. In this article we
use this second notion, which is sometimes used in the literature
(see \cite{J_L,Art-Llib-Schl}).

Polynomial vector fields can be compactified using different
techniques which give a global view of the phase portraits including
the trajectories close to infinity which leads to the notion of
infinite singular points (for  more details see, for example
\cite{Llibre-Schlomiuk:2004}).


Finite and infinite singular points may either be real or complex.
Most of the times one only needs to observe the real ones. We point
out that the sum of the multiplicities of all singular points of a
quadratic system (with a finite number of singular points) is always
$7$. The sum of the multiplicities of the infinite singular points
is always at least $3$, more precisely it is always $3$ plus the sum
of the multiplicities of the finite points which have gone to
infinity.

We use here the following terminology for singularities grouped in
the families:

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

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

\item We call \emph{nilpotent} a singular point with its eigenvalues
zero but its Jacobian matrix is not identically zero.

\item We call \emph{intricate} a singular point with its
Jacobian matrix identically zero.

This notation (except ``nilpotent'') was  proposed by Dana Schlomiuk
in a personal communication in order to avoid intersection with
previous well-known notations. We are grateful to her for the help.
\end{itemize}

We say that two points are \emph{Jordan relatives} if they both
belong to one of the families above.

Roughly speaking a singular point $p$ of an analytic differential
system $\chi$ is a \emph{multiple singularity of multiplicity
$m$} if $p$ produces $m$ singularities, as closed to $p$ as we
wish, in analytic perturbations $\chi_{\epsilon}$ of this system
and $m$ is the maximum such number. In polynomial differential
systems of fixed degree $n$ we have several possibilities for
obtaining multiple singularities.
\begin{itemize}
\item[(i)] A finite singular point can split into several finite
singularities in n-degree polynomial perturbations.

\item[(ii)] An infinite singular point could split into some
finite and some infinite singularities in n-degree polynomial
perturbations.

\item[(iii)]  $n$-degree perturbations of an infinite singularity
produce only infinite singular points of the systems.
\end{itemize}
To all these cases we can give a precise mathematical meaning
using the notion of intersection multiplicity at a point $p$ of
two algebraic curves.


Two foci (or saddles) are \emph{order equivalent} if their
corresponding orders coincide.

Semi-elemental saddle-nodes are always topologically equivalent.

 \begin{definition}  {\rm
Two singularities $p_1$ and $p_2$ of two polynomial vector
fields are \emph{congruently equivalent} if and only if they are
topologically equivalent, they have the same multiplicity, they
are Jordan relatives and in case of foci or saddles they are order
equivalent.}
\end{definition}

In this work we discuss the behavior of quadratic vector fields
globally around their singularities.

\begin{definition} {\rm
Let $\chi_1$ and $\chi_2$ be two polynomial vector fields each
having a finite number of singularities. We say that $\chi_1$ and
$\chi_2$ have \emph{congruent  equivalent configurations of
singularities} if and only if we have a bijection $\vartheta$
carrying the singularities of $\chi_1$ to singularities of
$\chi_2$ and for every singularity p of $\chi_1$, $\vartheta(p)$
is congruently equivalent with $p$.}
\end{definition}

\section{The notation for singular points}\label{sec:Notation}

In this section we present the notation that we use for describing
the singular points. The complete notation for singular points will
appear in our project of classification of finite and infinite
singular points of all $QS$.

This notation used here for describing finite and infinite singular
points of quadratic systems, can easily be extended to general
polynomial systems.

We start by distinguishing the finite and infinite singularities
denoting the first ones with lower case letters and the second
with capital letters. When describing in a row both finite and
infinite singular points, we will always order them first finite,
latter infinite with a semicolon (`;') separating them.

Starting with elemental points, we use the letters `$s$',`$S$' for
``saddles''; `$n$', `$N$' for ``nodes''; `$f$' for ``foci'' and
`$c$' for ``centers''.

An elemental singular point is called a \emph{weak singularity}
if the trace of its Jacobian is zero. It follows easily that such
a singular point could be either a focus or a center or a saddle.
In order to determine the stability of weak focus one needs to
compute higher order terms of a certain function (see \cite{Li}).
Depending on the number of the terms of this function  which
vanish we can determine the order of the focus. A similar
technique can be used also in the case of a weak saddle.

 Finite elemental foci (or saddles) are classified
according to their order as weak foci (or saddles). When the trace
of the Jacobian matrix evaluated at those singular points is not
zero, we call them strong saddles and strong foci and we maintain
the standard notations `$s$' and `$f$.' But when the trace is zero,
it is known that for quadratic systems they may have up to 3 orders
plus an integrable one, which corresponds to infinite order. So,
from the order 1 to order 3 we denote them by `$s^{(i)}$' and
`$f^{(i)}$' where $i=1,2,3$ is the order. For the integrable case,
the saddle remains a topological saddle and it will be denoted by
`$\is$'. In the second case we have a change in the topology of the
local phase portrait which makes the singular point a center and it
is denoted by `$c$'.

Foci and centers cannot appear as isolated singular points at
infinity and hence it is not necessary to introduce their order in
this case. In case of saddles, we can have weak saddles at infinity
but it is premature at this stage to describe them since the maximum
order of weak singularities in cubic systems is not yet known.

All non--elemental singular points are multiple points, in the sense
that when we perturb them within a nearby system they could split in
at least two elemental points. For finite singular points we denote
with a subindex their multiplicity as in `$\overline{s}_{(5)}$' or
in `$\widehat{es}_{(3)}$' (the meaning of the
`$\overline{\phantom{s}}$' and the `$\widehat{\phantom{s}}$' will be
explained below). The \emph{ multiplicity} of a singularity of a
$QS$ is the maximum number of singular points which can appear from
this singularity when we perturb it inside the class of all $QS$. In
order to describe the various kinds of multiplicity of infinite
singular points we use the concepts and notations introduced in
\cite{Dana-Vlp-JDE}. Thus we denote by `${a\choose b}...$' the
maximum number $a$ (respectively $b$) of finite (respectively
infinite) singularities which can be obtained by perturbation of the
multiple point. For example `$\overline{ {1\choose 1}  }\ SN$' means
a saddle--node at infinity produced by the collision of one finite
singularity with an infinite one; `$\overline{ {0\choose 3}  }\ S$'
means a saddle produced by the collision of 3 infinite
singularities.

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


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

It is known that the neighborhood of any singular point of a
polynomial vector field (except foci and centers) is formed by a
finite number of sectors which could only be of three types:
parabolic, hyperbolic and elliptic (see \cite{Dum-Llib-Art}). Then a
reasonable way to describe intricate points and nilpotent points at
infinity is to use a sequence formed by the types of their sectors.
The description we give is the one which appears in the clock--wise
direction once the blow--down is done. Thus in quadratic systems we
have just seven possibilities for finite intricate singular points
(see \cite{Art-Llib-Vlp-IJBCh}) which are the following ones
\begin{itemize}
\item{(a)} $phpphp_{(4)}$;
\item{(b)} $phph_{(4)}$;
\item{(c)} $hh_{(4)}$;
\item{(d)} $hhhhhh_{(4)}$;
\item{(e)} $peppep_{(4)}$;
\item{(f)} $pepe_{(4)}$;
\item{(g)} $ee_{(4)}$.
\end{itemize}

We use lower case because of the finite nature of the singularities
and add the subindex (4) since they are of multiplicity $4$.

For infinite intricate and nilpotent singular points, we insert a
hyphen between the sectors to split those which appear on one side
of the equator of the sphere from the ones which appear in the
other side. In this way we distinguish between
${2\choose2}PHP$-$PHP$ and ${2\choose 2}PPH$-$PPH$.

The lack of finite singular points will be encapsulated in the
notation \ses. In the cases we need to point out the lack of an
infinite singular point we will use the symbol $\emptyset$.

Finally there is also the possibility that we have an infinite
number of finite or infinite singular points. In the first case,
this means that the polynomials defining the differential system are
not coprime. Their common factor may produce a real line or conic
filled up with singular points, or a conic with real coefficients
having only complex points.

We consider now systems which have the set of non isolated
singularites located on the line at infinity.  It is known that
the neighborhood of infinity can be of $6$ different types (see
\cite{Dana-Vlp-JDE}) up to topological equivalence. The way to
determine them comes from a study of the reduced system on the
infinite local charts where the line of singularities can be
removed within the chart and still a singular point may remain on
the line at infinity.  Thus, depending of the nature of this
point, the behavior of the singularities at infinity of the
original system can be denoted as $[\infty, \emptyset]$,
$[\infty, N]$,  $[\infty, S]$, $[\infty, C]$,
$[\infty, \overline{SN}]$ or $[\infty, \widehat{ES}]$. In the
families showed in this paper we will only meet the case
$[\infty, S]$.

We will denote with the symbol $\boldsymbol{\ominus}$ the case when
the polynomials defining the system have a common factor. The symbol
stands for the most generic of these cases which corresponds to a
real line filled up of singular points. The degeneracy can be also be
produced by a common quadratic factor which could generate any kind
of conic. We will indicate each case by the following symbols
\begin{itemize}\label{pageref:notat}
\item  $\boldsymbol{\ominus}[|]$ for a real straight line;
\item  $\boldsymbol{\ominus}[\cup]$ for a real parabola;
\item  $\boldsymbol{\ominus}[\parallel]$ for two real parallel lines;
\item  $\boldsymbol{\ominus}[\parallel^c]$ for two complex parallel lines;
\item  $\boldsymbol{\ominus}[\boldsymbol{|}\text{\scriptsize 2}]$ for a double real straight line;
\item  $\boldsymbol{\ominus}[ \;\text{\footnotesize)(}\,]$ for a real hyperbola;
\item  $\boldsymbol{\ominus}[\times]$ for two intersecting real straight lines;
\item  $\boldsymbol{\ominus}[\circ]$ for a real circle or ellipse;
\item  $\boldsymbol{\ominus}[\copyright]$ for a complex conic;
\item  $\boldsymbol{\ominus}[\cdot]$ for two complex straight
lines which intersect at a real finite
point.
\end{itemize}

The cases  that will be considered in this paper are a subset of
the previous cases.

Moreover we also want to determine whether after removing the common
factor of the polynomials, singular points remain on the curve
defined by this common factor. If the reduced system has no finite
singularity which remains on the curve defined by this common
factor, we will use the symbol \ses to describe this situation. If
some singular points remain we will use the corresponding notation
of their types.

The existence of a common factor of the polynomials defining the
differential system also affects the infinite singular points. We
point out that the projective completion of a real affine line
filled up with singular points has a point on the line at infinity
which will then be also a non isolated singularity.

In order to describe correctly the singularities at infinity, we
must mention also this kind of phenomena and describe what happens
to such points at infinity after the removal of the common factor.
To show the existence of the common factor we will use the same
symbol as before: $\boldsymbol{\ominus}$, and for the type of
degeneracy we use the symbols introduced above. We will use the
symbol $\emptyset$ to denote the non-existence of infinite
singular points after the removal of the degeneracy. There are
other possibilities for a polynomial system, but this is the only
one of interest in this paper.


\section{Some preliminary results}\label{sec:preliminary}

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


\subsection{Number and types of weak singularities of quadratic
systems}\label{sec:weak}

A complete characterization of the finite weak singularities of
quadratic systems via invariant theory was done in
\cite{Vlp_prepr_B-2010}, where the   next result is proved.

\begin{proposition}\label{prop:weak-sing}
Consider a non-degenerate quadratic system \eqref{sys:QSgen}.
\begin{itemize}
\item[(a)] If  $\mathcal{T}_4\ne0$ then  this system has
no weak singularity.

\item[(b)] If  $\mathcal{T}_4=0$ and $\mathcal{T}_3\ne0$
then the system has  exactly one weak singularity. Moreover this
singularity is either a weak focus (respectively a weak saddle) of
the indicated order below,  or a center (respectively an integrable
saddle)  if and only if {$\mathcal{T}_3\mathcal{F}<0$} (respectively
{$\mathcal{T}_3\mathcal{F}>0$}) and the following corresponding
condition holds
\begin{itemize}
\item[($b_1$)]   $f^{(1)}$ (respectively  $s^{(1)}$)
$\Leftrightarrow$    $\mathcal{F}_1\ne0$;
\item[($b_2$)]    $f^{(2)}$ (respectively $s^{(2)}$)
$\Leftrightarrow$   $\mathcal{F}_1= 0$, $\mathcal{F}_2\ne0$;
\item[($b_3$)]   $f^{(3)}$   (respectively  $s^{(3)}$)
$\Leftrightarrow$ $\mathcal{F}_1= \mathcal{F}_2=0$,
 $\mathcal{F}_3\mathcal{F}_4\ne0$;
\item[($b_4$)]  $c$ (respectively \is)
$\Leftrightarrow$
$\mathcal{F}_1=\mathcal{F}_2=\mathcal{F}_3\mathcal{F}_4=0$.
\end{itemize}

\item[(c)] If  {$\mathcal{T}_4=\mathcal{T}_3=0$} and
{$\mathcal{T}_2\ne0$}, then the system could possess two and only two
weak singularities and none of them is of order $2$ or $3$. Moreover
this system possesses two weak singularities, which are of the types
indicated below, if and only if $\mathcal{F}=0$ and one of the
following conditions holds
\begin{itemize}
\item[($c_1$)] $s^{(1)},  s^{(1)}$ $\Leftrightarrow$
 $\mathcal{F}_1\ne0$, $\mathcal{T}_2<0$, $\mathcal{B}\le0$,
 $\mathcal{H}>0$;
\item[($c_2$)]  $s^{(1)},  f^{(1)}$ $\Leftrightarrow$
$\mathcal{F}_1\ne0$, $\mathcal{T}_2>0$, $\mathcal{B}<0$;
\item[($c_3$)]  $f^{(1)}, f^{(1)}$ $\Leftrightarrow$
$ \mathcal{F}_1\ne0$, $\mathcal{T}_2<0$, $\mathcal{B}<0$,
$\mathcal{H}<0$;
\item[($c_4$)]   $\is, \is$  $\Leftrightarrow$ $\mathcal{F}_1=0$,
$\mathcal{T}_2<0$, $\mathcal{B}<0$, $\mathcal{H}>0$;
\item[($c_5$)]  $\is ,  c$ $\Leftrightarrow$   $\mathcal{F}_1=0$,
$\mathcal{T}_2>0$, $\mathcal{B}<0$;
\item[($c_6$)]  $c,  c$  $\Leftrightarrow$ $\mathcal{F}_1=0$,
$\mathcal{T}_2<0$, $\mathcal{B}<0$, $\mathcal{H}<0$.
\end{itemize}

\item[(d)] If  {$\mathcal{T}_4=\mathcal{T}_3=\mathcal{T}_2=0$} and
{$\mathcal{T}_1\ne0$}, then the system could possess one and only one
weak singularity (which is of order 1). Moreover this system has one
weak singularity  of the type indicated below if and only if
$\mathcal{F}=0$ and one of the following conditions holds
\begin{itemize}
\item[($d_1$)]     $s^{(1)}$  $\Leftrightarrow$
$\mathcal{F}_1\ne0$, $\mathcal{B}<0$, $\mathcal{H}>0$;
\item[($d_2$)]  $f^{(1)}$   $\Leftrightarrow$  $\mathcal{F}_1\ne0$,
 $\mathcal{B}<0$, $\mathcal{H}<0$.
\end{itemize}

\item[(e)] If  {$\mathcal{T}_4=\mathcal{T}_3=
\mathcal{T}_2=\mathcal{T}_1=0$} and {$\sigma(a,x,y)\ne0$}, then  the
system could possess one and only one weak singularity. Moreover
this system has one  weak singularity, which is of the type
indicated below, if and only if one of the following conditions
holds
\begin{itemize}
\item[($e_1$)]  $s^{(1)}$ $\Leftrightarrow$  $\mathcal{F}_1\ne0$,
 $\mathcal{H}={\mathcal B_1}=0$, $\mathcal{B}_2>0$;
\item[($e_2$)]  $f^{(1)}$  $\Leftrightarrow$
$\mathcal{F}_1\ne0$,  $\mathcal{H}={\mathcal B_1}=0$,
$\mathcal{B}_2<0$;
\item[($e_3$)]
\[
\is \Leftrightarrow  \begin{cases} [\alpha]\; \mathcal{F}_1=0,\;
\mathcal{F}=0,\;
  \mathcal{B}<0,\; \mathcal{H}>0,\text{ or}\\
[\beta]\; \mathcal{F}_1=0,\;  \mathcal{H}={\mathcal B_1}=0,\;
 \mathcal{B}_2>0,\text{ or}\\
[\gamma]\; \mathcal{F}_1=0,\;
\mathcal{H}=\mathcal{B}={\mathcal B_1}=  \mathcal{B}_2
= \mathcal{B}_3=\mu_0=0,\; K(\mu_2^2+\mu_3^2)\ne0,\text{ or} \\
[\delta]\; \mathcal{F}_1=0,\;
   \mathcal{H}=\mathcal{B}={\mathcal B_1}=  \mathcal{B}_2
= \mathcal{B}_3=K=0,\  \mu_2{\mathcal G}\ne0,\text{ or}  \\
[\varepsilon]\; \mathcal{F}_1=0,\;
\mathcal{H}=\mathcal{B}={\mathcal B_1}= \mathcal{B}_2
= \mathcal{B}_3=\mathcal{B}_4=K=\mu_2=0,\; \mu_3\ne0;
\end{cases}
\]

\item[($e_4$)]
\[
 c    \Leftrightarrow \begin{cases}
[\alpha]\; \mathcal{F}_1=0,\; \mathcal{F}=0,\;
   \mathcal{B}<0,\; \mathcal{H}<0,\text{ or}\\
[\beta]\; \mathcal{F}_1=0,\;   \mathcal{H}={\mathcal B_1}=0,\;
 \mathcal{B}_2<0.
\end{cases}
\]
\end{itemize}

\item[(f)] If $\sigma(\tilde a,x,y)=0$, then the system is Hamiltonian and
it  possesses $i$ (with $1\le i\le 4$) weak singular points   of the
types indicated below if and only if one of the following conditions
holds
\begin{itemize}
\item[($f_1$)]  $\is ,\is ,\is ,c$   $\Leftrightarrow$ $\mu_0<0$,
 $\mathbf{D}<0$,  $\mathbf{R}>0$, $\mathbf{S}>0$;
\item[($f_2$)]  $\is ,\is , c, c$ $\Leftrightarrow$  $\mu_0>0$,
$\mathbf{D}<0$, $\mathbf{R}>0$, $\mathbf{S}>0$;
\item[($f_3$)]  $\is , \is , c$  $\Leftrightarrow$  $\mu_0=0$,
$\mathbf{D}<0$, $\mathbf{R}\ne0$;
\item[($f_4$)]
\[
\is ,\is \Leftrightarrow \begin{cases}
[\alpha]\; \mu_0<0,\; \mathbf{D}>0,\text{ or}\\
[\beta]\;  \mu_0<0,\; \mathbf{D}=0, \; \mathbf{T}<0,\text{ or}\\
[\gamma]\;  \mu_0=\mathbf{R}=0,\; \mathbf{P}\ne0, \;
\mathbf{U}>0,\; K\ne0;
\end{cases}
\]
\item[($f_5$)]
\[
  \is , c   \Leftrightarrow \begin{cases}
[\alpha]\; \mu_0>0,\; \mathbf{D}>0,\text{ or}\\
[\beta]\;   \mu_0>0,\; \mathbf{D}=0, \; \mathbf{T}<0,\text{ or}\\
[\gamma]\;   \mu_0=\mathbf{R}=0,\; \mathbf{P}\ne0, \;
\mathbf{U}>0,\; K=0;
\end{cases}\]

\item[($f_6$)]
\[
 \is  \Leftrightarrow \begin{cases}
[\alpha]\; \mu_0<0,\; \mathbf{D}=\mathbf{T}=\mathbf{P}=0, \;
 \mathbf{R}\ne0,\text{ or}\\
[\beta]\;   \mu_0=0,\; \mathbf{D}>0, \; \mathbf{R}\ne0,\text{ or}\\
[\gamma]\;   \mu_0=0,\; \mathbf{D}=0, \; \mathbf{P}\mathbf{R}\ne0,
\text{ or}\\
[\delta]\;   \mu_0=\mathbf{R}=\mathbf{P}=0, \; \mathbf{U}\ne0;
\end{cases}
\]

\item[($f_7$)]  $c$  $\Leftrightarrow$ $\mu_0>0$,
$\mathbf{D}=\mathbf{T}=\mathbf{P}=0$, $\mathbf{R}\ne0$.
\end{itemize}
\end{itemize}
\end{proposition}

The invariant polynomials  used in the above theorem  are constructed
as follows
%\label{Pol:F_i_1}
\begin{gather*}
\mathcal{F}_1(\tilde a)=A_2,\\
\begin{aligned}
\mathcal{F}_2(\tilde a)
&=-2 A_1^2 A_3 + 2 A_5 (5 A_8 + 3 A_9) +
 A_3 (A_8 - 3 A_{10} + 3 A_{11} + A_{12}) \\
 &\quad - A_4 (10 A_8 - 3 A_9 + 5 A_{10} + 5A_{11} + 5A_{12}),
\end{aligned}\\
\begin{aligned}
\mathcal{F}_3(\tilde a)
&= -10 A_1^2 A_3 + 2 A_5 (A_8 -A_9) -
 A_4 (2 A_8 + A_9 + A_{10} + A_{11} + A_{12})   \\
&\quad + A_3 (5 A_8 + A_{10} - A_{11} + 5 A_{12}),
\end{aligned}\\
\begin{aligned}
\mathcal{F}_4(\tilde a)
&= 20 A_1^2 A_2 - A_2 (7 A_8 - 4
A_9 + A_{10} + A_{11} + 7 A_{12}) +
 A_1 (6 A_{14} - 22 A_{15}) \\
&\quad - 4 A_{33} + 4 A_{34},
\end{aligned}
\\
\mathcal{F}(\tilde a)= A_7,\quad
\mathcal{B}(\tilde a)= -(3 A_8 + 2 A_9 + A_{10} + A_{11}
+ A_{12}),\\
\mathcal{H}(\tilde a)=-(A_4+2A_5),\quad
\mathcal{G}(\tilde a,x,y)= \widetilde M+32\widetilde H,
\end{gather*}
and
%\label{Pol:B1,B2}
\begin{gather*}
\begin{aligned}
\mathcal{B}_1(\tilde a)
&=\Big\{\big(T_7,D_2\big)^{(1)}\big[12 D_1 T_3+2D_1^3+9D_1T_4
 +36\big(T_1,D_2\big)^{(1)}\big]
 - 2D_1\big(T_6,D_2\big)^{(1)}\\
&\quad\times \big[D_1^2 + 12T_3]
+D_1^2\big[D_1\big(T_8,C_1\big)^{(2)}
+  6\big(\big(T_6,C_1\big)^{(1)}, D_2\big)^{(1)}\big]\Big\}/144,
\end{aligned}
\\ %\label{Pol:B3,B4}
\begin{aligned}
\mathcal{B}_2(\tilde a)
&=\Big\{\big(T_7,D_2\big)^{(1)}\big[8T_3\big(T_6,D_2\big)^{(1)}-
D_1^2\big(T_8,C_1\big)^{(2)}-
          4D_1\big( \big(T_6,C_1\big)^{(1)},D_2\big)^{(1)}\big]\\
&\quad +\Big[\big(T_7,D_2\big)^{(1)}\Big]^2
(8T_3-3T_4+2D_1^2)\Big\}/384,
\end{aligned} \\
\mathcal{B}_3(\tilde a,x,y)= -D_1^2(4D_2^2+T_8+4T_9)
+3D_1D_2(T_6+4T_7)-24T_3(D_2^2-T_9),\\
\mathcal{B}_4(\tilde a,x,y)= D_1(T_5+2D_2C_1)-3C_2(D_1^2+2T_3).
\end{gather*}
Here by $(f,g)^{(k)}$ is denoted the differential operator
called \emph{transvectant of index $k$} (see
 \cite{Olver}) of two polynomials $f,$ $g\in$
$\mathbb{R}[\tilde  a,x,y]$
$$
  (f,g)^{(k)}=
   \sum_{h=0}^k (-1)^h {k\choose h}
   \frac{\partial^k f}{\partial x^{k-h}\partial y^h}\
   \frac{\partial^k g}{\partial x^h\partial y^{k-h}},
$$
and  $A_i(\tilde a)$ are the elements of the   minimal polynomial
basis of affine invariants up to degree 12 (containing 42 elements)
constructed in \cite{Bul-Tim}.  We have applied here only the
following elements (keeping the notation from \cite{Bul-Tim})
\begin{gather*}
  A_1 = \tilde A, \quad
 A_2 = (C_2, D)^{(3)}/12,\\
  A_3 = \big(\big(  (C_2, D_2)^{(1)},  D_2\big)^{(1)}, D_2
\big)^{(1)}/48,\quad
 A_4= (\widetilde H, \widetilde H)^{(2)},\\
  A_5= (\widetilde H, \widetilde K)^{(2)}/2,\quad
 A_6= (\widetilde E, \widetilde H)^{(2)}/2,\\
  A_7= \big( (C_2, \widetilde E)^{(2)}, D_2\big)^{(1)}/8,\quad
 A_8= \big( (\widetilde D, \widetilde H)^{(2)}, D_2\big)^{(1)}/8,\\
  A_9= \big(\big( (\widetilde D, D_2)^{(1)}, D_2\big)^{(1)},
D_2\big)^{(1)}/48, \quad
 A_{10} = \big( (\widetilde D, \widetilde K)^{(2)}, D_2\big)^{(1)}/8,\\
  A_{11}= (\widetilde F, \widetilde K)^{(2)}/4, \quad
 A_{12}= (\widetilde F, \widetilde H)^{(2)}/4,\\
 A_{14}= (\widetilde B, C_2)^{(3)}/36, \quad
 A_{15}= (\widetilde E, \widetilde F)^{(2)}/4,\\
  A_{33}= \big(\big(\big( (\widetilde D, D_2)^{(1)}, \widetilde F\big)^{(1)},
 D_2\big)^{(1)},
  D_2\big)^{(1)}/128,\\
 A_{34}= \big(\big(\big( (\widetilde D, \widetilde D)^{(2)}, D_2\big)^{(1)},
 \widetilde K\big)^{(1)}, D_2\big)^{(1)}/64,
\end{gather*}
where
\begin{gather*}
\tilde A=  \left(C_1,T_8-2T_9+D_2^2\right)^{(2)}/144,\\
\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_1D_2
   -T_5)\\
&\quad -9D_1^2C_2\Big]/36,
\end{aligned}\\
\widetilde E=
\Big[D_1(2T_9-T_8)-3\left(C_1,T_9\right)^{(1)}
      -D_2(3T_7+D_1D_2)\Big]/72,
\\
\begin{aligned}
\widetilde F
&=\Big[6D_1^2(D_2^2-4T_9)+4D_1D_2(T_6+6T_7)
    + 48C_0\left(D_2,T_9\right)^{(1)} -9D_2^2T_4
    + 288 D_1\widetilde E\\
 &\quad -24(C_2,\widetilde D)^{(2)} + 120(D_2,\widetilde D)^{(1)}
    - 36C_1\left(D_2,T_7\right)^{(1)}
    + 8D_1\left(D_2,T_5\right)^{(1)}\Big]/144,
\end{aligned}\\
\begin{aligned}
\widetilde B&= \Big\{16D_1\left(D_2,T_8\right)^{(1)}
 \left(3C_1D_1-2C_0D_2 +4T_2\right)
  +32C_0\left(D_2, T_9\right)^{(1)}\big(3D_1D_2\\
 &\quad -5T_6+9T_7\big) +2\left(D_2,  T_9\right)^{(1)}
 \big(27C_1T_4  -18C_1D_1^2  -32D_1T_2
 +32\left(C_0, T_5\right)^{(1)}\big)
\\
 &\quad +6\left(D_2, T_7\right)^{(1)}
 [8C_0(T_8-12T_9) -12C_1(D_1D_2+T_7)+ D_1(26C_2D_1+32T_5)\\
 &\quad  + C_2(9T_4+96T_3)] +6\left(D_2, T_6\right)^{(1)}
  [32C_0T_9-C_1(12T_7+52D_1D_2) -32C_2D_1^2]\\
&\quad +48D_2\left(D_2, T_1\right)^{(1)} \left(2D_2^2-T_8\right)
  -32D_1T_8\left(D_2, T_2\right)^{(1)}+9D_2^2T_4
 \left(T_6-2T_7\right)\\
 &\quad -16D_1\left(C_2, T_8\right)^{(1)}\left(D_1^2+4T_3 \right)
 +12D_1\left(C_1, T_8\right)^{(2)}\left(C_1D_2-2C_2D_1 \right)\\
 &\quad +6D_1D_2T_4\left(T_8-7D_2^2-42T_9\right)
 +12D_1\left(C_1,T_8\right)^{(1)}\left(T_7+2D_1D_2\right)\\
&\quad  +96D_2^2\left[D_1\left(C_1, T_6\right)^{(1)}
 +D_2\left(C_0, T_6\right)^{(1)}\right]
 -16D_1D_2T_3\left(2D_2^2+3T_8\right)\\
&\quad  -4D_1^3D_2\left(D_2^2+3T_8+6T_9\right)
  +6D_1^2D_2^2\left(7T_6+2T_7\right)
  -252D_1D_2T_4T_9\Big\} /(2^{8}3^{3}),
\end{aligned}\\
\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, \\
\widetilde M= (C_2,C_2)^{(2)}= 2 \operatorname{Hess}\big(C_2(x,y)\big)
\end{gather*}
and
\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}
are the $GL$-comitants constructed by using   the following
five polynomials, basic ingredients in constructing invariant
polynomials for systems \eqref{sys:QSgen}
\begin{equation}\label{expr:Ci,Di} \begin{gathered}
   C_i(\tilde a,x,y)=yp_i(x,y)-xq_i(x,y),\ (i=0,1,2),\\
   D_i(\tilde a,x,y)=\frac{\partial p_i}{\partial x}+
                        \frac{\partial q_i}{\partial y},\ (i=1,2).
\end{gathered}
\end{equation}
The affine invariants $\mathcal{T}_j$ $(j=1,2,3,4)$ which are
responsible for the number of vanishing traces  of the finite
singularities (see \cite{Vlp_prepr_B-2010}) are constructed as
follows.

We  consider the polynomial  $\sigma(\tilde a,x,y)$ which  is an
affine comitant of systems \eqref{sys:QSgen}
$$
\sigma(\tilde a,x,y)=\frac{\partial  P}{\partial x}+\frac{\partial
Q}{\partial y}= \sigma_0(\tilde  a)+\sigma_1(\tilde  a,x,y)\ (\equiv
D_1(\tilde  a)+ D_2(\tilde  a,x,y)),
$$
and the differential operator $\mathcal{L}= x\cdot \mathbf{L}_2
-y\cdot\mathbf{L}_1$  (see \cite{Blt-Vlp-BASM}) acting on
$\mathbb{R}[\tilde a,x,y]$, where
\begin{gather*}
 \mathbf{L}_1= 2a_{00}\frac{\partial}{\partial a_{10}} +
            a_{10}\frac{\partial}{\partial a_{20}} +
    \frac{1}{2}a_{01}\frac{\partial}{\partial a_{11}} +
            2b_{00}\frac{\partial}{\partial b_{10}} +
            b_{10}\frac{\partial}{\partial b_{20}} +
     \frac{1}{2}b_{01}\frac{\partial}{\partial b_{11}};\\
\mathbf{L}_2= 2a_{00}\frac{\partial}{\partial a_{01}} +
            a_{01}\frac{\partial}{\partial a_{02}} +
     \frac{1}{2}a_{10}\frac{\partial}{\partial a_{11}} +
            2b_{00}\frac{\partial}{\partial b_{01}} +
            b_{01}\frac{\partial}{\partial b_{02}} +
     \frac{1}{2}b_{10}\frac{\partial}{\partial b_{11}}.
\end{gather*}
Applying the differential operators $\mathcal{L}$  and
$(*,*)^{(k)}$ (i.e. transvectant of index $k$) in
\cite{Vlp_prepr_B-2010} is defined the following polynomial function
(named \emph{trace  function})
\begin{equation}\label{fuct:T1st}
\mathfrak{T}
(w)=\sum_{i=0}^4\frac{1}{(i!)^2}\Big(\sigma_1^i, \frac{1}{i!}
\mathcal{L}^{(i)}(\mu_0)\Big)^{(i)}\,w^{4-i}=\sum_{i=0}^4
\mathcal{G}_i \,w^{4-i},
 \end{equation}
where the coefficients
$\mathcal{G}_i(\tilde  a)=
\frac{1}{(i!)^2}(\sigma_1^i, \mu_i)^{(i)}$,
 $i=0,1,2,3,4$ $\big(\mathcal{G}_0(\tilde  a)\equiv\mu_0(\tilde  a)\big)$
are $GL$-invariants.

Finally using the function $\mathfrak{T}(w)$  the following four
needed  affine invariants $\mathcal{T}_4$,  $\mathcal{T}_3$,
$\mathcal{T}_2$, $\mathcal{T}_1$ are constructed
\cite{Vlp_prepr_B-2010}
$$
 { \mathcal{T}_{4-i}(\tilde a)  } =\frac{1}{i!}\,
\frac{d^i { \mathfrak{T}  }}{dw^i}\Big|_{w=\sigma_0},\quad
 i=0,1,2,3\quad  \big(\mathcal{T}_4\equiv \mathfrak{T}(\sigma_0)\big),
$$
which are basic schematic affine invariants for the characterization
of weak singularities via invariant polynomials (see Proposition
\ref{prop:weak-sing}).

The  invariant polynomials
$\mathbf{D}, \mathbf{P}, \mathbf{R}, \mathbf{S}, \mathbf{T}, \mathbf{U}$
and $\mathbf{V}$ are defined in the next section (see
\eqref{Pol:D,T,R,e.al.}).

In what follows we also need the following  invariant polynomials:
\begin{equation}\label{Expr:Bj}
\begin{gathered}
B_3(\tilde a,x,y)=(C_2,\widetilde D)^{(1)}
 =\operatorname{Jacob}\left( C_2,\widetilde D\right),\\
B_2(\tilde a,x,y)= \left(B_3,B_3\right)^{(2)} - 6B_3(C_2,\widetilde D)^{(3)},\\
B_1(\tilde a)=\operatorname{Res}_x\left( C_2,\widetilde D\right)/y^9
=-2^{-9}3^{-8}\left(B_2,B_3\right)^{(4)},\\
 B_4(\tilde a,x,y)=-\big((\widetilde D,\widetilde H)^{(2)},
\widetilde H\big)^{(1)}\times(\widetilde D,\widetilde H)^{(2)},\\
B_5(\tilde a,x,y)=D_2\big[((C_2,D_2)^{(1)},D_2)^{(1)}-3(C_2,
\widetilde K)^{(2)}\big] ,\\
B_6(\tilde a,x,y)= C_1^2-4C_0C_2.
\end{gathered}
\end{equation}
and
\begin{equation}\label{Expr:M,N,R...}
\begin{gathered}
\eta(\tilde a) = {\rm Discrim}\big(C_2(x,y)\big)=(\widetilde M,\widetilde M)^{(2)}/384,\\
 \widetilde N(\tilde a,x,y)=  4\widetilde K-4 \widetilde H; \ \
  \widetilde R(\tilde a,x,y)=  \tilde L+32 \widetilde K, \\
 \theta(\tilde a) = {\rm Discrim}\big( \widetilde N(\tilde a,x,y)\big)=-(\widetilde N,\widetilde N)^{(2)}/2,\\
\tilde L(\tilde a,x,y)   = 16\widetilde K-32\widetilde
H-\widetilde M;\ \
\theta_1(\tilde a)    =  16\eta-2\theta-16\mu_0. \\
\end{gathered}
\end{equation}


\subsection{Number and multiplicities of the finite singularities
of quadratic systems}

The conditions for the number and multiplicities of the finite
singularities of quadratic systems were first  constructed in
\cite{Blt-Vlp-BASM}.

We  shall use here the notion of \emph{zero--cycle} in order to
describe the number and multiplicity of singular points of a
quadratic system. This notion as well as   the notion of
\emph{divisor}, were used for classification purposes of planar
quadratic differential systems by Pal and Schlomiuk
\cite{Pal-Schlomiuk:2001a},  Llibre and
Schlomiuk \cite{Llibre-Schlomiuk:2004},
Schlomiuk and Vulpe \cite{Dana-Vlp-JDE}
and by Artes and Llibre and Schlomiuk~\cite{Art-Llib-Schl}.

\begin{definition} \label{def:zero-cycle} {\rm
We consider formal expressions
$\mathcal{D} = \sum n(w)w$\ where $n(w)$ is an integer and only a
finite number of $n(w)$ are nonzero. Such an expression is called
a \emph{zero--cycle of $\mathbf{P}_2(\mathbb{C})$} if all $w$ appearing  in
$\mathcal{D}$ are points of $\mathbf{P}_2(\mathbb{C})$. We call \emph{degree}
of the zero--cycle $\mathcal{D}$ the integer $\deg(\mathcal{D}) =
\sum n(w)$. We call \emph{support} of   $\mathcal{D}$ the set
$\operatorname{supp}(\mathcal{D})$ of  $w$'s appearing in $\mathcal{D}$ such
that $n(w)\ne0$.
}\end{definition}

We note that $\mathbf{P}_2(\mathbb{C})$ denotes the complex projective
space of dimension 2. For a system $(S)$ belonging to the family
\eqref{sys:QSgen} we denote
$\nu(P,Q)= \{w\in\mathbb{C}_2: P(w)=Q(w)=0\}$ and  we define
the  zero-cycle
$\mathcal{D}_{{}_S}(P,Q)  = \sum_{w\in\nu(P,Q)}I_w(P,Q)w$, where
$I_w(P,Q)$ is the intersection number or multiplicity of intersection
of $P$ and $Q$ at $w$.  It is clear that for a non--degenerate
quadratic system $\deg(\mathcal{D}_{{}_S})\le4$ as well as
$\operatorname{supp}(\mathcal{D}_{{}_S})\le4$. For a degenerate system  the
zero--cycle $\mathcal{D}_{{}_S}(P,Q)$ is undefined.

Using the  affine invariant polynomials
\begin{equation}\label{expr: mu0,DD,TT,..}
 \mu_0(\tilde a),\quad
\mathbf{D}(\tilde a),\quad \mathbf{R}(\tilde a,x,y),\quad
\mathbf{S}(\tilde a,x,y),\quad
\mathbf{T}(\tilde a,x,y),\quad  \mathbf{U}(\tilde a,x,y),\quad
\mathbf{V}(\tilde a,x,y)
\end{equation}
(the construction of these polynomials will be discussed
further), in \cite{Blt-Vlp-BASM} the next proposition was proved.

\begin{proposition}[\cite{Blt-Vlp-BASM}] \label{prop:number-FSPs}
The form of the divisor $ \mathcal{D}_{{}_S}(P,Q)$ for non-degenerate
quadratic systems \eqref{sys:QSgen} is determined by the
corresponding conditions indicated in Table \ref{table3}, where we write
$p+q+r^c+s^c$ if two of the finite points, i.e. $r^c, s^c$, are
complex but not real.
\end{proposition}

\begin{table}[ht] \scriptsize
\caption{} \label{table1}
\begin{center}
\begin{tabular}{|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|}
\hline
  \,No. & {\begin{tabular}{c} Zero--cycle\\ $ \mathcal{D}_{{}_S}(P,Q)$\end{tabular}}
  &   \begin{tabular}{c}Invariant \\ criteria \end{tabular}
  & \,No.
  &  {\begin{tabular}{c} Zero--cycle\\ $ \mathcal{D}_{{}_S}(P,Q)$\end{tabular}}
  &   \begin{tabular}{c}Invariant \\ criteria \end{tabular}  \\
\hline
 1 & $p+q+r+s$ &$\begin{array}{c} \mu_0\ne0, \mathbf{D}<0,\\[-0.1mm] \mathbf{R}>0, \mathbf{S}>0\end{array}$&
  10 & \,$p+q+r $\, &$\begin{array}{c} \mu_0=0, \mathbf{D}<0, \mathbf{R}\ne0\end{array}$\\
\hline
 2 & \,$p+q+r^c+s^c$\, &$\begin{array}{c} \mu_0\ne0, \mathbf{D}>0\end{array}$&
 11 & $p+q^c+r^c $ &$\begin{array}{c} \mu_0=0, \mathbf{D}>0, \mathbf{R}\ne0\end{array}$\\
\hline
 \raisebox{-0.7em}[0pt][0pt]{3}
 & \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}p^{\,c}+q^c+r^c+s^c\end{array}$}
 &$\begin{array}{c} \mu_0\ne0, \mathbf{D}<0, \mathbf{R}\le0\end{array}$ &
 \raisebox{-0.7em}[0pt][0pt]{12} & \raisebox{-0.7em}[0pt][0pt]{$2p+q$} &
   \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c} \mu_0=\mathbf{D}=0, \mathbf{P}\mathbf{R}\ne0\end{array}$}\\
 \cline{3-3}
  &  & $\begin{array}{c} \mu_0\ne0, \mathbf{D}<0, \mathbf{S}\le0\end{array}$ &    & &\\
\hline
   {4} & {$2p+q+r $} & {$\begin{array}{c}  \mu_0\ne0, \mathbf{D}=0, \mathbf{T}<0 \end{array}$}&
  13 &  $3p $  &$\begin{array}{c} \mu_0 = \mathbf{D} = \mathbf{P} = 0, \mathbf{R}\ne0\end{array}$\\
\hline
 5 & $2p+q^c+r^c$ &$\begin{array}{c}  \mu_0\ne0, \mathbf{D}=0, \mathbf{T}>0\end{array}$&
 14 & $p+q$ &$\begin{array}{c} \mu_0=\mathbf{R}=0, \mathbf{P}\ne0,\\[-0.1mm]\mathbf{U}>0\end{array}$\\
\hline
 6 & \,$2p+2q $\, &$\begin{array}{c} \mu_0\ne0, \mathbf{D}=\mathbf{T}=0,\\[-0.1mm] \mathbf{P}\mathbf{R}>0\end{array}$ &
 15 & \,$p^{\,c}+q^c $\, &$\begin{array}{c} \mu_0=\mathbf{R}=0, \mathbf{P}\ne0,\\[-0.1mm]\mathbf{U}<0\end{array}$\\
\hline
 7 & $2p^{\,c}+2q^c$ &$\begin{array}{c}   \mu_0\ne0, \mathbf{D}=\mathbf{T}=0,\\[-0.1mm] \mathbf{P}\mathbf{R}<0\end{array}$&
 16 & $2p $ &$\begin{array}{c} \mu_0=\mathbf{R}=0, \mathbf{P}\ne0,\\[-0.1mm]\mathbf{U}=0\end{array}$\\
\hline
 8 & \,$3p+q $\, &$\begin{array}{c} \mu_0\ne0, \mathbf{D}=\mathbf{T}=0,\\[-0.1mm]\mathbf{P}=0,\mathbf{R}\ne0\end{array}$ &
 17 & \,$p $\, &$\begin{array}{c} \mu_0=\mathbf{R}=  \mathbf{P}=0,\\[-0.1mm] \mathbf{U}\ne0 \end{array}$\\
\hline
 9 & $4p $ &$\begin{array}{c}  \mu_0\ne0, \mathbf{D}=\mathbf{T}=0,\\[-0.1mm] \mathbf{P}=\mathbf{R}=0\end{array}$&
 18 & $0$ &$\begin{array}{c} \mu_0=\mathbf{R}=\mathbf{P}=0,\\[-0.1mm]\mathbf{U}=0, \mathbf{V}\ne0\end{array}$\\
\hline
\end{tabular}
\end{center}
\end{table}


\section{The global diagram for the finite singularities of
quadratic systems. Some needed invariants} \label{sec:diagram}

We note that the polynomials \eqref{expr: mu0,DD,TT,..} were
constructed in \cite{Blt-Vlp-BASM} (see also
\cite{Art-Llib-Vlp-IJBCh}) using the basic ingredients
\eqref{expr:Ci,Di} in constructing invariant polynomials  for
systems \eqref{sys:QSgen} and applying the  differential operator
$(*,*)^{(k)}$ (i.e. transvectant of index $k$).


Here we shall use  the new expressions for the polynomials
\eqref{expr: mu0,DD,TT,..} (constructed in \cite{Vlp_prepr_B-2010}),
which are equivalent to the old ones but make more transparent their
geometry and allow us to observe the dynamic of the finite
singularities. More exactly we shall use the polynomials
$\mu_0(\tilde a)$ and $\mu_i(\tilde a,x,y)$ constructed in
\cite{Blt-Vlp-BASM} as follows
\begin{equation}\label{expr:mu:i}
\begin{gathered}
\mu_0(\tilde a) =  \operatorname{Res}_x\big(p_2(x,y),q_2(x,y)\big)/y^4, \\
\mu_i(\tilde a,x,y) = \frac{1}{i!} \mathcal{L}^{(i)}(\mu_0), \quad
 i=1,\dots ,4,
\end{gathered}
\end{equation}
where $ \mathcal{L}^{(i)}(\mu_0)
=\mathcal{L}(\mathcal{L}^{(i-1)}(\mu_0))$.
Their geometrical meaning is revealed in the following two lemmas.

\begin{lemma}[\cite{Blt-Vlp-BASM}] \label{lem:mu_i-Degen}
The total multiplicity of all finite singularities of a quadratic
system \eqref{sys:QSgen} equals $k$ if and only if for every
$i\in\{0,1,\dots,k-1\}$ we have $\mu_i(\tilde a,x,y)=0$  in
$\mathbb{R}[x,y]$   and $\mu_k(\tilde a,x,y)\ne0$. Moreover a system
\eqref{sys:QSgen} is degenerate if and only if $\mu_i(\tilde
a,x,y)=0$ in $\mathbb{R}[x,y]$ for every $i=0,1,2,3,4$.
\end{lemma}


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

Using the invariant polynomials $\mu_i$ $(i=0,1,\dots,4)$ in
\cite{Vlp_prepr_B-2010} the polynomials \eqref{expr: mu0,DD,TT,..}
are constructed as follows
\begin{equation}\label{Pol:D,T,R,e.al.}
\begin{aligned}
\mathbf{D}&= \Big[3\big((\mu_3,\mu_3)^{(2)}, \mu_2\big)^{(2)}- \big(6\mu_0\mu_4-3\mu_1\mu_3+\mu_2^2,\ \mu_4)^{(4)}\Big]/48,\\
\mathbf{P}&= 12\mu_0\mu_4-3\mu_1\mu_3+\mu_2^2,\\
\mathbf{R}&= 3\mu_1^2-8\mu_0\mu_2,\\
\mathbf{S}&= \mathbf{R}^2-16\mu_0^2\mathbf{P},\\
\mathbf{T}&= 18\mu_0^2(3\mu_3^2-8\mu_2\mu_4)+2\mu_0(2\mu_2^3-9\mu_1\mu_2\mu_3+27\mu_1^2\mu_4)-
               \mathbf{P}\mathbf{R},\\
\mathbf{U}&= \mu_3^2-4\mu_2\mu_4,\\
\mathbf{V}&=  \mu_4.\\
\end{aligned}
\end{equation}
Considering these  expressions we have  the next remark.

\begin{remark}  \label{rem: muj-equiv-R,P,U,V-0} \rm
If $\mu_0=0$ then the condition
$\mathbf{R}=0$ (respectively $\mathbf{R}=\mathbf{P}=0$;
$\mathbf{R}=\mathbf{P}=\mathbf{U}=0$;
$\mathbf{R}=\mathbf{P}=\mathbf{U}=\mathbf{V}=0$)
is equivalent to $\mu_1=0$ (respectively
$\mu_1=\mu_2=0$; $\mu_1=\mu_2=\mu_3=0$;
$\mu_1=\mu_2=\mu_3=\mu_4=0$).
\end{remark}

On the other hand, considering  Lemma \ref{lem:mu_i-Degen} we
deduce that the invariant polynomials $\mu_i$ $(i=0,1,\dots,4)$
are responsible for the number of finite singularities which have
coalesced  with infinite ones. So taking into account the remark
above and  Proposition \ref{prop:number-FSPs} we could present a
diagram, which is equivalent to Table 1. So we get the next
result.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig1}
%\includegraphics[width=0.8\textwidth]{EPS-files/Diagr_Finite-Mltipl-Total-B.eps}
\end{center}
\caption{Diagram for Finite Singularities of Quadratic systems}
\label{fig1}
\end{figure}


\begin{theorem}\label{thm:diagram-FSPs}
The number and multiplicities of the
finite singular points (described by the divisor $
\mathcal{D}_{S}(P,Q)$) for   non-degenerate quadratic systems
\eqref{sys:QSgen} is given by the diagram presented in
 Figure \ref{fig1}.
\end{theorem}


We are interested in a global characterization  of the singularities
(finite and infinite) of the family of quadratic systems. More
precisely we would like to extend the diagram of Figure \ref{fig1}
adding the
infinite singularities (their number and multiplicities) and then
including the types of all these singularities. Moreover we wish to
distinguish the weak singularities (if it is the case) as well as
their order. This is one of the motivations  why we consider again
the class of quadratic systems with centers  as well as the class of
Hamiltonian systems, the topological classifications of which could
be found in articles \cite{KalSib,Kal90,Kal92} and \cite{KalVul},
respectively.

On the other hand we would like to reveal the main affine invariant
polynomials associated to the singularities of quadratic systems,
having a transparent geometrical meaning. And it is clear  that all
the conditions we need have to be based on the invariant polynomials
contained in the diagram of Figure \ref{fig1}.

Thus in this article new geometrical more transparent affine
invariant conditions for distinguishing topological phase portraits
of the  two mentioned families of quadratic systems are
simultaneously constructed. For this purpose  we need the following
$GL$-invariant polynomials constructed  in tensorial form in
\cite{Sib88-Mon} (we  keep the respective notations)
\begin{equation}\label{inv-com:tens-form}
\begin{gathered}
I_1= a^\alpha_\alpha,\quad
I_2 = a^\alpha_\beta a^\beta_\alpha,\quad
I_3 =  a^\alpha_pa^\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_7 =  a^\alpha_{pr} a^\beta_{\alpha q}
  a^\gamma_{\beta s}a^\delta_{\gamma \delta}
  \varepsilon^{pq}\varepsilon^{rs}, \\
I_8 = a^\alpha_{pr} a^\beta_{\alpha q}
  a^\gamma_{\delta s}a^\delta_{\beta \gamma}
 \varepsilon^{pq}\varepsilon^{rs},\quad
I_9 =  a^\alpha_{pr} a^\beta_{\beta q}
 a^\gamma_{\gamma s}a^\delta_{\beta \gamma}
 \varepsilon^{pq}\varepsilon^{rs},\\
I_{10} = a^\alpha_{p} a^\beta_{\delta}
  a^\gamma_{\nu}a^\delta_{\alpha q} a^{\nu}_{\beta\gamma}
  \varepsilon^{pq},\quad
I_{13} = a^\alpha_p a^\beta_{q r}a^\gamma_{\gamma s}a^\delta_{\alpha
  \beta}a^\nu_{\delta \nu} \varepsilon^{pq}\varepsilon^{rs},\\
I_{16} =  a^\alpha_p a^\beta_{r}a^\gamma_{\delta}
 a^\delta_{\alpha q}a^\nu_{\beta s}a^\tau_{\gamma\mu}
 a^\mu_{\nu\tau} \varepsilon^{pq}\varepsilon^{rs},\quad
I_{18} = a^\alpha a^q  a^p_\alpha \varepsilon_{pq}, \\
I_{19} = a^\alpha a^\beta_\gamma a^\gamma_{\alpha \beta},\quad
I_{21} = a^\alpha a^\beta a^q a^p_{\alpha \beta} \varepsilon_{pq},\quad
I_{23} = a^\alpha a^\beta a^\gamma_{\alpha \delta}
 a^\delta_{\beta \gamma} ,\\
I_{24} = a^\alpha  a^\beta_\delta a^\gamma_\alpha
 a^\delta_{\beta \gamma},\quad
I_{28} =  a^\alpha a^\beta  a^\gamma_\delta
a^\delta_{\gamma \mu}a^\mu_{\alpha \beta} ,\quad
I_{30} = a^\alpha a^\beta_p a^\gamma_{\beta q}
 a^\delta_{\gamma \mu}a^\mu_{\alpha \delta}\varepsilon^{pq},
\end{gathered}
\end{equation}

 \begin{equation}\label{inv-com:tens-form-1}
\begin{gathered}
I_{33} = a^\alpha a^\beta a^\gamma a^\delta_{\alpha \beta}
    a^\mu_{\gamma \nu}a^\nu_{\delta \mu} ,\quad
I_{35} =  a^\alpha  a^\beta_p a^\gamma_\alpha
  a^\delta_{\beta q}a^\mu_{\gamma \nu}a^\nu_{\delta \mu}
  \varepsilon^{pq},\\
K_1 =  a^\alpha_{\alpha \beta}  x^\beta,\quad
K_2 = a^p_\alpha x^\alpha x^q \varepsilon_{pq} ,\quad
K_3 = a^\alpha_\beta a^\beta_{\alpha \gamma} x^\gamma,\quad
K_5 = a^p_{\alpha \beta} x^\alpha x^\beta x^q \varepsilon_{pq},\\
K_7 = a^\alpha_{\beta\gamma} a^\beta_{\alpha\delta} x^\gamma
x^\delta
 \quad
K_{11} = a^p_\alpha a^\alpha_{\beta\gamma} x^\beta x^\gamma x^q
 \varepsilon_{pq}, \quad
K_{12} =  a^\alpha_\beta  a^\beta_{\alpha\gamma}
 a^\gamma_{\delta\mu} x^\delta x^\mu,\\
K_{14} =  a^\alpha_p a^\beta_{\alpha q} a^\gamma_{\beta\delta}
 a^\delta_{\gamma\mu} x^\mu\varepsilon^{pq} ,\quad
K_{21} = a^p x^q \varepsilon_{pq} ,\quad
K_{22} = a^\alpha a^p_\alpha x^q \varepsilon_{pq},\\
K_{23} =  a^p a^q_{\alpha \beta} x^\alpha x^\beta \varepsilon_{pq},\quad
K_{27} = a^\alpha a^\beta_{\alpha\gamma} a^\gamma_{\beta \delta}
  x^\delta,\quad
K_{31} = a^\alpha  a^\beta_{\alpha\gamma} a^\gamma_{\beta\delta}
 a^\delta_{\mu\nu} x^\mu x^\nu,
\end{gathered}
\end{equation}
where $ \varepsilon_{11} = \varepsilon_{22} =
\varepsilon^{11} = \varepsilon^{22} = 0 $,
$ \varepsilon_{12} =
-\varepsilon_{21}=\varepsilon^{12} = -\varepsilon^{21} = 1 $.
We note that  the expressions for the above invariants are associated
to the tensor notation for   quadratic systems \eqref{sys:QSgen}
(see \cite{Sib88-Mon})
\begin{gather*}
   \frac{dx^j}{ dt} =a^{j}+ a^{j}_{\alpha}x^{\alpha}
+ a^{j}_{\alpha\beta}   x^{\alpha}x^{\beta}, \quad
(j,\alpha, \beta =1,2); \\
 a^{1}= a_{00}, \quad 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}= b_{00}, \quad 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{gather*}

\section{The family of quadratic systems with
centers}\label{sec:centers}

\begin{table}[ht] \scriptsize
\caption{} \label{table2}
\begin{center}
\begin{tabular}{|@{}c@{}|@{}c@{} |@{} c @{}|@{} c @{}|  c |  c  |}

 \hhline{|======|} \multicolumn{6}{|c|}{\begin{tabular}{c}\emph{Conditions for the existence of
 a center} [statement $(b)$ of Proposition \ref{prop:weak-sing}]:\\
 $\mathcal{T}_3\ne0$, $\mathcal{T}_4=\mathcal{F}_1=
  \mathcal{F}_2=\mathcal{F}_3\mathcal{F}_4=0$, $\mathcal{T}_3\mathcal{F}<0$,\ $(b_4)$ \end{tabular}}   \rule{0pt}{6.7mm}\\[0.3mm]
 \hhline{|======} \multicolumn{4}{|c|}{\begin{tabular}{c}\emph{Additional   conditions for configurations} \end{tabular}} &
 \begin{tabular}{c}\emph{Configuration}\\[-1mm] \emph{of singularities}\end{tabular} & No. \rule{0pt}{6.7mm}\\[0.3mm]
\hline
  \multirow{5}*{\raisebox{-0.5em}[0pt][0pt]{ $\begin{array}{c}\mu_0\ne0\end{array}$}}
  &  \multirow{1}*{\raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}\mathbf{D}<0\end{array}$}}
   & \multicolumn{2}{|c|}{$\begin{array}{c}\widetilde K<0\end{array}$} & $c, s, s, s\,;\  N,  N,   N$ &1\rule{0pt}{4.0mm}\\[0.3mm]
\cline{3-6}
   &   & \multicolumn{2}{|c|}{$\begin{array}{c}\widetilde K>0\end{array}$} & $c, s, n, n\,;\  S,  S, \, N$ &2\rule{0pt}{4.0mm}\\[0.3mm]
\cline{2-6}
   &  \multirow{1}*{\raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}\mathbf{D}>0\end{array}$}}
   & \multicolumn{2}{|c|}{$\begin{array}{c}\eta<0\end{array}$} & $c, f\,;\  S $ &3\rule{0pt}{3.5mm}\\
\cline{3-6}
   &   & \multicolumn{2}{|c|}{$\begin{array}{c}\eta>0\end{array}$} & $c, n\,;\, S, S, \, N$ &4\rule{0pt}{3.5mm}\\
\cline{2-6}
   &\multicolumn{3}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\mathbf{D}=0\end{array}$}}}
  & $c,  n,\  \overline{ sn}_{(2)}\,;\,  S,\overline{ {0\choose 2}  }\
SN$ & 5\rule{0pt}{4.5mm}\\[0.3mm]
\hline
   \multicolumn{4}{|c|}{\multirow{1}*{\raisebox{-0.0em}[0pt][0pt]{ $\begin{array}{c}\mu_0=0\end{array}$}}}
  & $c ,  s\,;\, \, N,  \overline{ {1\choose 1}  }\ SN,  \overline{ {1\choose 1}  }\
SN$ &6\rule{0pt}{4.5mm}\\[0.3mm]
\hhline{|======|} \multicolumn{6}{|c|}{\begin{tabular}{c}\emph{Conditions
for the existence of a center}  [statement $(c)$ of Proposition \ref{prop:weak-sing}]:\\
 $\mathcal{T}_4=\mathcal{T}_3=0$, $\mathcal{T}_2\ne0$,
$(c_5)\cup(c_6)$ \end{tabular}}   \rule{0pt}{6.7mm}\\[0.3mm]
 \hline
    \multicolumn{1}{|@{}c }{\multirow{7}*{\raisebox{-0.3em}[0pt][0pt]{ $\begin{array}{c}\mu_0<0\end{array}$}}}
  &\multicolumn{1}{|c|}{\multirow{4}*{ \raisebox{-0.4em}[0pt][0pt]{$\begin{array}{c}\mathbf{D}<0\end{array}$}}}
  &\multicolumn{2}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\eta<0\end{array}$}}}
  &  $c,\is ,n,n\,;\ S $ &7\rule{0pt}{3.5mm}\\[0.3mm]
\cline{3-6} &
  &\multirow{2}*{ \raisebox{-0.3em}[0pt][0pt]{$\begin{array}{c}\eta>0\end{array}$}}&
  \multicolumn{1}{|c|}{\multirow{1}*{
  \raisebox{-0.1em}[0pt][0pt]{$\begin{array}{c}\widetilde K<0\end{array}$}}}
 & $c,\is,s,s\,;\   N,  N, N$ & 8\rule{0pt}{4.0mm}\\[0.3mm]
\cline{4-6} & & &
  \multicolumn{1}{|c|}{\multirow{1}*{
  \raisebox{-0.1em}[0pt][0pt]{$\begin{array}{c}\widetilde K>0\end{array}$}}}
 & $c,\is ,n,n\,;\ S, S, N$ &9\rule{0pt}{4.0mm}\\[0.3mm]
\cline{3-6} &
  &\multicolumn{2}{|c|}{\multirow{1}*{
  \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\eta=0\end{array}$}}}
   & $c,\is ,n,n\,;\ \overline{ {0\choose 3}  }\
S  $&10\rule{0pt}{4.3mm}\\[0.3mm]
 \cline{2-6}
   &\multicolumn{1}{|c|}{\multirow{3}*{ \raisebox{-0.2em}[0pt][0pt]{$\begin{array}{c}\mathbf{D}>0\end{array}$}}}
  &\multicolumn{2}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\eta<0\end{array}$}}}
  &  $c, c\,; \ S $ &11\rule{0pt}{3.5mm}\\[0.3mm]
\cline{3-6}
   & &\multicolumn{2}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\eta>0\end{array}$}}}
  &  $c, c\,; \ S,  S, N  $ &12\rule{0pt}{3.5mm}\\[0.3mm]
\cline{3-6} &
  &\multicolumn{2}{|c|}{\multirow{1}*{
  \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\eta=0\end{array}$}}}
   & $c, c\,; \ \overline{ {0\choose 3}  }\ S  $&13\rule{0pt}{4.3mm}\\[0.3mm]
 \hline
    \multicolumn{1}{|@{}c }{\multirow{4}*{\raisebox{-0.3em}[0pt][0pt]{ $\begin{array}{c}\mu_0>0\end{array}$}}}
  &\multicolumn{3}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\mathbf{D}<0\end{array}$}}}
    & $c ,c ,s ,s\,;\    N $&14\rule{0pt}{3.5mm}\\[0.3mm]
 \cline{2-6}
  &\multicolumn{1}{|c|}{\multirow{3}*{ \raisebox{-0.2em}[0pt][0pt]{$\begin{array}{c}\mathbf{D}>0\end{array}$}}}
  &\multicolumn{2}{|c|}{$\begin{array}{c}\eta<0\end{array}$}  & $c, \is\,;  \  N $ &15\rule{0pt}{3.5mm}\\[0.3mm]
\cline{3-6}
  &\multicolumn{1}{|c|}{}
  &\multicolumn{2}{|c|}{$\begin{array}{c}\eta>0\end{array}$}  & $c, \is\,;  \  S,  N,  N $ &16\rule{0pt}{3.5mm}\\[0.3mm]
\cline{3-6}
  &\multicolumn{1}{|c|}{}
  &\multicolumn{2}{|c|}{$\begin{array}{c}\eta=0\end{array}$}  & $c, \is\,;\ \overline{ {0\choose 3}  }\ N $
  & 17\rule{0pt}{4.5mm}\\[0.3mm]
 \hline
  \multicolumn{1}{|@{}c }{\multirow{5}*{\raisebox{-0.5em}[0pt][0pt]{ $\begin{array}{c}\mu_0=0\end{array}$}}}
  &\multicolumn{1}{|c|}{\multirow{3}*{ \raisebox{-0.3em}[0pt][0pt]{$\begin{array}{c}\eta<0\end{array}$}}}
  &\multicolumn{1}{|c|}{\multirow{1}*{ \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}\widetilde K\ne0\end{array}$}}}
   &$\begin{array}{c}\mu_2<0\end{array}$ & $c, c\,; \ \overline{ {2\choose 1}  }\ S$ & 18\rule{0pt}{4.5mm}\\[0.3mm]
 \cline{4-6}
 \multicolumn{1}{|@{}c }{ }
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{}
   &$\begin{array}{c}\mu_2>0\end{array}$ & $c, \is \,;\ \overline{ {2\choose 1}  }\ N$ & 19\rule{0pt}{4.5mm}\\[0.3mm]
 \cline{3-6}
   &\multicolumn{1}{|c|}{}
   &\multicolumn{2}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\widetilde K=0\end{array}$}}}
   & $c, \is\,;\, N$ & 20\rule{0pt}{4.0mm}\\[0.3mm]
 \cline{2-6}
  \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{\multirow{1}*{ \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}\eta>0\end{array}$}}}
  &\multicolumn{2}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\widetilde K\ne0\end{array}$}}}
  & $c ,\is \,; \ \overline{ {2\choose 1}  }\ S, N, N $& 21\rule{0pt}{4.5mm}\\[0.3mm]
\cline{3-6}
   &\multicolumn{1}{|c|}{}
   &\multicolumn{2}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\widetilde K=0\end{array}$}}}
    & $c, \is \,;\, N ,\overline{ {1\choose 1}  }\ SN,  \overline{ {1\choose 1}  }\ SN $& 22\rule{0pt}{4.5mm}\\[0.3mm]
\hline
\end{tabular}
\end{center}
\end{table}


\begin{table}[ht] \scriptsize
\begin{center}
\begin{tabular}{|@{}c@{}|@{}c@{} |@{} c @{}|@{} c @{}|@{} c @{}|@{} c @{}| c | c |}
\multicolumn{7}{r}{Table \ref{table2}}(\emph{continued})\\
 \hhline{|=======|} \multicolumn{7}{|c|}{\begin{tabular}{c}\emph{Conditions for the existence of a center} [statement $(e)$ of Proposition \ref{prop:weak-sing}]:\\
 $\mathcal{T}_4=\mathcal{T}_3= \mathcal{T}_2=0,$ $\sigma\ne0$,
$(e_4)$\end{tabular}}   \rule{0pt}{6.7mm}\\[0.3mm]
 \hhline{|=======} \multicolumn{5}{|c|}{\begin{tabular}{c}\emph{Additional   conditions for configurations} \end{tabular}} &
 \begin{tabular}{c}\emph{Configuration}\\[-1mm] \emph{of singularities}\end{tabular} &\ No.  \rule{0pt}{5.0mm}\\[0.3mm]
\hline
   \multicolumn{1}{|@{}c }{\multirow{3}*{\raisebox{-0.2em}[0pt][0pt]{ $\begin{array}{c}\mu_0<0\end{array}$}}}
  &\multicolumn{4}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\eta<0\end{array}$}}}
     & $c, \widehat{es}_{(3)}\,;\  S $ & 23\rule{0pt}{3.5mm}\\[0.3mm]
\cline{2-7}
   \multicolumn{1}{|@{}c }{}
   &\multicolumn{4}{|c|}{\multirow{1}*{ \raisebox{-0.1em}[0pt][0pt]{$\begin{array}{c}\eta>0\end{array}$}}}
   & $c,  \widehat{es}_{(3)}\,;\ S, S,  N $ &24\rule{0pt}{3.5mm}\\[0.3mm]
\cline{2-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{4}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\eta=0\end{array}$}}}
     & $c, \widehat{es}_{(3)}\,;\ \overline{ {0\choose 3}  }\ S$& 25\rule{0pt}{4.5mm}\\[0.3mm]
\hline
   \multicolumn{5}{|@{}c| }{\multirow{1}*{\raisebox{-0.0em}[0pt][0pt]{ $\begin{array}{c}\mu_0>0\end{array}$}}}
     & $c,   \overline{s}_{(3)}\,;\ N $ & 26\rule{0pt}{3.5mm}\\[0.3mm]
 \hline
  \multicolumn{1}{|@{}c }{\multirow{9}*{\raisebox{-0.0em}[0pt][0pt]{ $\begin{array}{c}\mu_0=0\end{array}$}}}
 &\multicolumn{1}{|c|}{\multirow{5}*{ \raisebox{-0.4em}[0pt][0pt]{$  \begin{array}{c}\mu_1\ne0\end{array}  $}}}
 &\multicolumn{3}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$  \begin{array}{c}\mathbf{D}<0\end{array}  $}}}
   &\ $c, s,  s\,;\   N,\widehat{1\choose 2} P\;E\;P - H\,$ &27\rule{0pt}{4.7mm}\\[0.3mm]
\cline{3-7}
  \multicolumn{1}{|@{}c }{ }
 &\multicolumn{1}{|c|}{ }
 &\multicolumn{1}{|c|}{\multirow{4}*{ \raisebox{-0.4em}[0pt][0pt]{$  \begin{array}{c}\mathbf{D}>0\end{array}  $}}}
 &\multicolumn{2}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$  \begin{array}{c}\widetilde N\le0 \end{array}  $}}}
    & $  c\,; \,  S, \widehat{1\choose 2} P\;E\;P - H $ &28\rule{0pt}{5.0mm}\\[0.3mm]
\cline{4-7}
  \multicolumn{1}{|@{}c }{ }
 &\multicolumn{1}{|c|}{ }
 &\multicolumn{1}{|c|}{ }
 &\multicolumn{1}{|c|}{\multirow{3}*{ \raisebox{-0.2em}[0pt][0pt]{$  \begin{array}{c}\widetilde N>0 \end{array}  $}}}
 &\multicolumn{1}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$  \begin{array}{c}\tilde L<0\end{array}  $}}}
     & $\  c\,; \,  S, \widehat{1\choose 2} P\;E\;P -  P\;H\;P$
     &29\rule{0pt}{5.2mm}\\[0.3mm]
\cline{5-7}
  \multicolumn{1}{|@{}c }{ }
 &\multicolumn{1}{|c|}{ }
 &\multicolumn{1}{|c|}{ }
 &\multicolumn{1}{|c|}{ }
 &\multicolumn{1}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$  \begin{array}{c}\tilde L>0\end{array}  $}}}
     & $\  c\,; \,   N, \widehat{1\choose 2}H -  H\;H\;H $
     &30\rule{0pt}{5.0mm}\\[0.3mm]
\cline{5-7}
  \multicolumn{1}{|@{}c }{ }
 &\multicolumn{1}{|c|}{ }
 &\multicolumn{1}{|c|}{ }
 &\multicolumn{1}{|c|}{ }
 &\multicolumn{1}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$  \begin{array}{c}\tilde L=0\end{array}  $}}}
     & $  c\,; \   [\infty, S] $    &31\rule{0pt}{4.0mm}\\[0.3mm]
\cline{2-7}
  \multicolumn{1}{|@{}c }{ }
 &\multicolumn{1}{|c|}{\multirow{3}*{ \raisebox{-0.2em}[0pt][0pt]{$  \begin{array}{c}\mu_1=0\end{array}  $}}}
 &\multicolumn{1}{|c|}{\multirow{1}*{ \raisebox{-0.7em}[0pt][0pt]{$  \begin{array}{c}\mu_3\ne0\end{array}  $}}}
 &\multicolumn{2}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$  \begin{array}{c}\tilde L\ne0\end{array}  $}}}
   &\ $c\,;\     N,{3\choose 2} H -  H\;H\;H $ &32\rule{0pt}{4.0mm}\\[0.3mm]
\cline{4-7}
  \multicolumn{1}{|@{}c }{ }
 &\multicolumn{1}{|c|}{ }
 &\multicolumn{1}{|c|}{ }
 &\multicolumn{2}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$  \begin{array}{c}\tilde L=0\end{array}  $}}}
   &\ $c\,;\   \overline{ {2\choose 1}  }\ {S},\widehat{1\choose 2} P\;E\;P -  H $ &33\rule{0pt}{5.0mm}\\[0.3mm]
\cline{3-7}
  \multicolumn{1}{|@{}c }{ }
 &\multicolumn{1}{|c|}{ }
 &\multicolumn{3}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$  \begin{array}{c}\mu_3=0\end{array}  $}}}
   &\ $c, \big(  \boldsymbol{\ominus}[|];\raisebox{0.2em}[0pt][0pt]{${}_\emptyset$}\big);\ \big(  \boldsymbol{\ominus}[|];\emptyset\big)$  &34\rule{0pt}{4.0mm}\\[0.3mm]
\hhline{|=======|} \multicolumn{7}{|c|}{
\begin{tabular}{c}\emph{Conditions
for the existence of a center}  [statement $(f)$ of Proposition \ref{prop:weak-sing}]:\\
 Hamiltonian systems \ $\Rightarrow$\ $\sigma=0$,
 $(f_1)$--$(f_3)$, $(f_5)$, $(f_7)$ \end{tabular}}   \rule{0pt}{6.7mm}\\[0.3mm]
 \hline
   \multicolumn{5}{|@{}c| }{\multirow{1}*{\raisebox{-0.2em}[0pt][0pt]{ $\begin{array}{c}\mu_0<0\end{array}$}}}
     & $c, \is ,\is ,\is \,; \  N, N, N$ & 35\rule{0pt}{3.5mm}\\[0.3mm]
  \hline
   \multicolumn{2}{|@{}c }{\multirow{4}*{\raisebox{-0.2em}[0pt][0pt]{ $\begin{array}{c}\mu_0>0\end{array}$}}}
  &\multicolumn{3}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\mathbf{D}<0\end{array}$}}}
     & $c ,c, \is ,\is \,;\   N$ &36\rule{0pt}{3.5mm}\\[0.3mm]
\cline{3-7}
   \multicolumn{2}{|@{}c }{}
  &\multicolumn{3}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\mathbf{D}>0\end{array}$}}}
     & $c,\is \,; \ N$ &37\rule{0pt}{3.5mm}\\[0.3mm]
  \cline{3-7}
   \multicolumn{2}{|@{}c }{}
  &\multicolumn{2}{|c|}{\multirow{2}*{ \raisebox{-0.2em}[0pt][0pt]{$\begin{array}{c}\mathbf{D}=0\end{array}$}}}
  &\multicolumn{1}{|c|}{$  \begin{array}{c}\mathbf{T}\ne0\end{array}  $}
     & $c,\is , \widehat{cp}_{(2)}\,;\  N $ &38\rule{0pt}{3.5mm}\\[0.3mm]
\cline{5-7}
   \multicolumn{2}{|@{}c }{}
  &\multicolumn{2}{|c|}{ }
  &\multicolumn{1}{|c|}{$  \begin{array}{c}\mathbf{T}=0\end{array}  $}
     & $c , \widehat s_{(3)}\,;\ N  $ &39\rule{0pt}{3.5mm}\\[0.3mm]
\hline
   \multicolumn{2}{|@{}c }{\multirow{2}*{\raisebox{-0.1em}[0pt][0pt]{ $\begin{array}{c}\mu_0=0\end{array}$}}}
  &\multicolumn{3}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\mu_1\ne0\end{array}$}}}
     & $c, \is ,\is \,; \ N, \widehat{1\choose 2}PEP-H$ &40\rule{0pt}{5.0mm}\\[0.3mm]
\cline{3-7}
   \multicolumn{2}{|@{}c }{}
  &\multicolumn{3}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\mu_1=0\end{array}$}}}
    & $c,\is \,; \ \widehat{2\choose 3}\, N $& 41\rule{0pt}{5.0mm}\\[0.3mm]
 \hline
\end{tabular}
 \end{center}
\end{table}

\begin{table}[ht] \scriptsize
\caption{} \label{table3}
\begin{center}
\begin{tabular}{|@{}c@{ }| c ||@{}c@{}|@{} c@{}||@{}c@{}|@{}c@{}|}
 \hhline{|======|}
{  \begin{tabular}{c}\emph{Con-}\\[-0.7mm] \emph{figu-}\\[-0.7mm] \emph{ration} \end{tabular}  }&
  {  \begin{tabular}{c}\emph{Phase}\\[-0.7mm] \emph{portrait} \end{tabular}  } &
{  \begin{tabular}{c}\emph{Con-}\\[-0.7mm] \emph{figu-}\\[-0.7mm] \emph{ration} \end{tabular}  }&
 {  \begin{tabular}{c}\emph{Phase}\\[-0.7mm] \emph{portrait} \end{tabular}  } &
{  \begin{tabular}{c}\emph{Con-}\\[-0.7mm] \emph{figu-}\\[-0.7mm] \emph{ration} \end{tabular}  }&
{  \begin{tabular}{c}\emph{Phase}\\[-0.7mm] \emph{portrait} \end{tabular}  }\rule{0pt}{1.7mm}\\[0.3mm]
 \hhline{|======|}
1 & \emph{Vul}$_{\,10}$ &  15  & \emph{Vul}$_{\,2}$ & 30 & \emph{Vul}$_{\,13}$  \rule{0pt}{3.6mm}\\[0.0mm]
 \hline
2 & \emph{Vul}$_{\,27}$ &  16  & \emph{Vul}$_{\,19}$ & 31 & \emph{Vul}$_{\,15}$  \rule{0pt}{3.6mm}\\[0.0mm]
 \hline
3 & \emph{Vul}$_{\,30}$ &  17  & \emph{Vul}$_{\,2}$ & 32 & \emph{Vul}$_{\,13}$  \rule{0pt}{3.6mm}\\[0.0mm]
 \hline
4 & \emph{Vul}$_{\,32}$ &  18  & \emph{Vul}$_{\,20}$ & 33 & \emph{Vul}$_{\,12}$  \rule{0pt}{3.6mm}\\[0.0mm]
 \hline
5 & \emph{Vul}$_{\,31}$ &  19  & \emph{Vul}$_{\,2}$ & 34 & \emph{Vul}$_{\,29}$  \rule{0pt}{3.6mm}\\[0.0mm]
 \hline
6 & \emph{Vul}$_{\,17}$ &  20  & \emph{Vul}$_{\,2}$ &  \raisebox{-2.2em}[0pt][0pt]{35}  & \emph{Vul}$_{\,11}$ if $B_1\ne0$  \rule{0pt}{3.6mm}\\[0.0mm]
\cline{1- 4}\cline{6-6}
7 & \emph{Vul}$_{\,25}$ &  21   & \emph{Vul}$_{\,19}$  &    & \emph{Vul}$_{\,9}$ if $B_1=0$, $B_3B_4<0$  \rule{0pt}{3.6mm}\\[0.0mm]
\cline{1- 4}\cline{6-6}
  & \emph{Vul}$_{\,9}$ if $B_3B_5<0$ &    & \emph{Vul}$_{\,18}$ if $B_3B_5<0$\, &    & \emph{Vul}$_{\,8}$ if $B_1=0$, $B_3B_4>0$ \rule{0pt}{3.6mm}\\[0.0mm]
 \cline{2- 2}\cline{4-4}\cline{6-6}
8 & \emph{Vul}$_{\,8}$ if $B_3B_5>0$&  22 & \emph{Vul}$_{\,16}$ if $B_3B_5>0$ &   &  \emph{Vul}$_{\,10}$ if $B_1= B_3=0$ \rule{0pt}{3.6mm}\\[0.0mm]
 \cline{2- 2}\cline{4- 6}
  & \emph{Vul}$_{\,10}$ if $B_3=0$&    &  \emph{Vul}$_{\,17}$ if $B_3=0$ &  \raisebox{-0.7em}[0pt][0pt]{36} & \emph{Vul}$_{\,4}$ if $B_1\ne0$  \rule{0pt}{3.6mm}\\[0.0mm]
\cline{1- 4}\cline{6-6}
  & \emph{Vul}$_{\,28}$ if $B_3B_5<0$ &  \raisebox{-0.7em}[0pt][0pt]{23} & \emph{Vul}$_{\,22}$ if $\theta_1<0$ &  & \emph{Vul}$_{\,3}$ if $B_1=0$  \rule{0pt}{3.6mm}\\[0.0mm]
 \cline{2- 2}\cline{4- 6}
9 & \emph{Vul}$_{\,26}$ if $B_3B_5>0$&    & \emph{Vul}$_{\,23}$ if $\theta_1\ge0$ & 37  &  \emph{Vul}$_{\,2}$ \rule{0pt}{3.6mm}\\[0.0mm]
 \cline{2-6}
  & \emph{Vul}$_{\,27}$ if $B_3=0$&  24 &  \emph{Vul}$_{\,24}$ & 38 & \emph{Vul}$_{\,7}$ \rule{0pt}{3.6mm}\\[0.0mm]
 \hline
10 & \emph{Vul}$_{\,25}$ &  25  & \emph{Vul}$_{\,23}$ & 39 & \emph{Vul}$_{\,2}$  \rule{0pt}{3.6mm}\\[0.0mm]
 \hline
11 & \emph{Vul}$_{\,20}$ &  26  & \emph{Vul}$_{\,2}$ & \raisebox{-0.7em}[0pt][0pt]{40} & \emph{Vul}$_{\,6}$ if $B_1\ne0$  \rule{0pt}{3.6mm}\\[0.0mm]
  \cline{1- 4}\cline{6-6}
12 & \emph{Vul}$_{\,21}$ & 27    &  \emph{Vul}$_{\,5}$ &  & \emph{Vul}$_{\,5}$ if $B_1=0$  \rule{0pt}{3.6mm}\\[0.0mm]
 \hline
13 & \emph{Vul}$_{\,20}$ & 28  & \emph{Vul}$_{\,12}$ & 41  & \emph{Vul}$_{\,2}$  \rule{0pt}{3.6mm}\\[0.0mm]
 \hline
14 & \emph{Vul}$_{\,3}$ &  29  & \emph{Vul}$_{\,14}$
&\multicolumn{2}{|@{}c }{}
\rule{0pt}{0.0mm}\\[0.0mm]
\cline{1-  4}
\end{tabular}
\end{center}
\end{table}

The proof of Theorem \ref{thm:QS-center} is based on the
classification of  the family of quadratic systems with centers given
in \cite{Vulpe-DU}. Using the expressions \eqref{inv-com:tens-form}
of \cite{Vulpe-DU} we get the following $GL$-invariants (we keep the
respective notations adding only  the "hat")
\begin{equation}\label{inv:alpha-delta}
\begin{gathered}
\hat\alpha=I_2^2I_8-28I_2I_5^2+6I_5I_{10},\quad
\hat\beta=4I_4^2-3 I_2I_9-4I_3I_4,\\
\hat\gamma=\frac{3}{I_4^2}(2I_3I_4+I_2I_9),\quad
\hat\delta=27I_8-I_9-18I_7,\\
\hat\xi=I_2I_5(I_2I_5+2I_{10})-4I_{10}^2-I_2^3I_8.
\end{gathered}
\end{equation}
We note that in \cite{Vulpe-DU}  the expressions of the
invariants $ \hat\xi$ and $\hat\delta$ are   used directly, but we
set these notations for compactness.

According to \cite{Vulpe-DU} (see Lemmas 2-5) the next result
follows.

\begin{proposition}\label{prop:GL-can-syst}
Assume that for a quadratic system with
the singular point $(0,0)$ the conditions $I_1=I_6=0$ and
$I_2<0$ hold. Then this system has a  center  at $(0,0)$ and via a
linear transformation could be brought to one of the canonical forms
below if and only if the respective additional $GL$-invariant
conditions hold
\begin{gather*}
 (S^{(c)}_1)\quad \begin{cases}
 \dot x=-y+g x^2-xy,\; (g\ne0),\\
 \dot  y=x+x^2+3g xy -2y^2,\end{cases}\\
 \Longleftrightarrow  I_3I_{13}\ne0,\
5I_3-2I_4=13I_3-10I_5=0;
\\
(S^{(c)}_2)\quad \begin{cases}
  \dot x= y+2n xy, \; (wm\ne0),\\
  \dot y=-x+lx^2+2m xy -ly^2, \end{cases}
  \Longleftrightarrow  I_3=0,\; I_{13}\ne0;
\\
(S^{(c)}_3)\quad \begin{cases}
 \dot x= y+2(1-e) xy,\\
 \dot y=-x+dx^2 +ey^2,\end{cases}
  \Longleftrightarrow   I_{13}=0,\;  I_4\ne0;
\\
(S^{(c)}_4)\quad \begin{cases}
 \dot x= y+2c xy+by^2,\\
 \dot y=-x-ax^2 -cy^2,\;  c\in\{0,1/2\} \end{cases}
 \Longleftrightarrow    I_4=0;
\end{gather*}
where $w=m^2 (2 n - l) - (n - l)^2 (2 n + l)$.
\end{proposition}

\emph{Proof of Theorem \ref{thm:QS-center}}. We shall consider
each one of the systems $(S^{(c)}_1)-(S^{(c)}_4)$ and  will compare
the $GL$-invariant conditions   \cite{Vulpe-DU}   with the affine
invariant ones given by Tables \ref{table2} and \ref{table3}.

\subsection{The family of  systems $(S^{(c)}_1)$} For these systems
we calculate the respective $GL$-invariants
\begin{equation}\label{val:S1-GL}
I_{13}=125 g (1 + g^2)/8,\quad  I_3=5(1 + g^2)/2
\end{equation}
and the affine invariant polynomials
\begin{equation}\label{val:S1-Aff}
\begin{gathered}
\mathcal{T}_4=\mathcal{F}_1=\mathcal{F}_2=\mathcal{F}_4=0,\quad
\mathcal{T}_3=-125 g (1 + g^2),\quad \mathcal{F}_3=625 g^2 (1 +
g^2)^2,\\ \mu_0=-2(1 + g^2),\quad \mathbf{D}=5184g^2 (1 + g^2),\quad
\eta=4(1 + g^2).
\end{gathered}
\end{equation}
According to \cite{Vulpe-DU} in the case
$I_{13}\ne0$ the phase portrait of systems $(S^{(c)}_1)$ is given
by \emph{Vul}$_{\,32}$ and hence we have the configuration of the
singularities $c, n;\, S, S, \, N$.

On the other hand the condition $I_{13}\ne0$ implies
$\mathcal{T}_3\ne0$ and as
$\mathcal{F}_1=\mathcal{F}_2=\mathcal{F}_3\mathcal{F}_4=0$, the
conditions provided by the statement $(b_4)$ of Proposition
\ref{prop:weak-sing} are satisfied. Moreover, as $\mu_0\ne0$,
$\mathbf{D}>0$ and $\eta>0$,  we obtain the respective conditions given by
Table \ref{table2} (row No. 4).

\subsection{The family of  systems $(S^{(c)}_2)$}

In this case calculations yield:
\begin{equation}\label{val:S2-GL}
\begin{gathered}
I_{13}=m \big[m^2 (2 n
- l) - (n - l)^2 (2 n + l)\big],\quad
I_9-I_8=4  l n ( l^2 + m^2 + n^2),\\
\hat\delta=-8 (l + 2 n)^2 (l^2 + m^2 + 2 l n).
 \end{gathered}
\end{equation}
According to \cite{Vulpe-DU} (see Lemma 4) the phase portrait
(and this yields the respective configuration of singularities) of a
system from the family $(S^{(c)}_2)$ is determined by the following
$GL$-invariant conditions, respectively
\begin{equation}\label{cond:ph-port-S2}
\begin{array}{lllll}
I_9-I_8>0 & \Leftrightarrow & \emph{Vul}_{10} &\Rightarrow & c,s,s,s; N,N,N;\\
 I_9-I_8=0 & \Leftrightarrow & \emph{Vul}_{17} &\Rightarrow & c ,  s\,;\, \, N,  \overline{ {1\choose 1}  }\ SN,  \overline{
{1\choose 1}  }\ SN;\\
 I_9-I_8<0,\ \hat\delta<0 & \Leftrightarrow & \emph{Vul}_{27} &\Rightarrow & c, s, n, n;   S,  S,   N;\\
 I_9-I_8<0,\ \hat\delta>0 & \Leftrightarrow & \emph{Vul}_{30} &\Rightarrow & c, f;   S;\\
 I_9-I_8<0,\ \hat\delta=0 & \Leftrightarrow & \emph{Vul}_{31} &\Rightarrow &
 c,  n, \overline{ sn}_{(2)};  N,\overline{0\choose 2}\
{SN}.\\
\end{array}
\end{equation}

On the other hand calculations yield
\begin{equation}\label{val:S2-Aff}
\begin{gathered}
\mathcal{T}_4=\mathcal{F}_1=\mathcal{F}_2=\mathcal{F}_3=0,\quad
\mathcal{T}_3\mathcal{F}=-8m^2 \big[m^2 (2 n - l)
 - (n - l)^2 (2 n + l)\big]^2, \\
\mu_0=-4 l^2 n^2,\quad \mathbf{D}=-192 (l + 2 n)^2 (l^2 + m^2 + 2 l
n)=-48\eta,\quad \widetilde K=-4ln(x^2+y^2).
\end{gathered}
\end{equation}
We observe that the condition $I_{13}\ne0$ implies
$\mathcal{T}_3\mathcal{F}<0$ and
as $\mathcal{F}_1=\mathcal{F}_2= \mathcal{F}_3\mathcal{F}_4=0$ we
conclude that the conditions provided by statement $(b_4)$ of
Proposition \ref{prop:weak-sing} are fulfilled. Moreover comparing
\eqref{val:S2-GL} and \eqref{val:S2-Aff} if $\mathbf{D}\mu_0\ne0$
we obtain
$$
\operatorname{sign}(I_9-I_8)
=-\operatorname{sign}(\widetilde K),\quad \operatorname{sign}(\hat\delta)=
\operatorname{sign}(\mathbf{D})
=- \operatorname{sign}(\eta),
$$
and $I_9-I_8=0$ (respectively $\hat\delta=0$) if and only if
$\mu_0=0$ (respectively $\mathbf{D}=0$). So taking into consideration
that the condition $\widetilde K<0$ implies $\mathbf{D}<0$,
we obviously arrive to the conditions provided by Table \ref{table2}
(the case of statement (b) of Proposition \ref{prop:weak-sing}).

\subsection{The family of  systems $(S^{(c)}_3)$}

For these systems calculations yield
\begin{equation}\label{val:S3-Ti}
\begin{gathered}
\mathcal{T}_4=\mathcal{T}_3=\mathcal{T}_1=\mathcal{F}=\mathcal{F}_1=0,
\quad
\mathcal{T}_2=4d(d+2-2e),\\
 \mathcal{B}=-2,\quad
\mathcal{H}=4d(1-e),\quad \sigma=2y.
\end{gathered}
\end{equation}
Therefore according to Proposition \ref{prop:weak-sing} for a
system $(S^{(c)}_3)$ could be satisfied either the conditions of
statement $(c)$ (if $\mathcal{T}_2\ne0$) or of statement $(e)$ (if
$\mathcal{T}_2=0$). We shall consider each one of these
possibilities.

\subsubsection{The case $\mathcal{T}_2\ne0$}

Following \cite[Lemma 3]{Vulpe-DU} for systems $(S^{(c)}_3)$ we
calculate
\begin{equation}\label{val:S3-GL}
\begin{gathered}
I_4=-1,\quad I_3=-(d+e),\quad
I_9 =d,\quad \hat\beta=2(d+2-2e),\quad
\hat\gamma=6e,
\end{gathered}
\end{equation}
and therefore the condition $\mathcal{T}_2\ne0$ is equivalent to
$I_9\hat\beta\ne0$. It was above mentioned that the conditions of
statement $(c)$ (Proposition \ref{prop:weak-sing}) are satisfied in
this case; i.e., there should be two weak singularities on the phase
plane of these systems. So according to \cite{Vulpe-DU} (see the
proof of Lemma 3) the phase portrait (and this yields the respective
configuration of singularities) of a system from the family
$(S^{(c)}_3)$ with  the condition $I_9\hat\beta\ne0$ is determined
by the following $GL$-invariant conditions
\begin{gather} \label{cond:ph-port-S3-1A}
\hat\gamma I_9>0,  \;  \hat\beta\hat\gamma<0
   \Leftrightarrow    \emph{Vul}_3  \Rightarrow  c,c,s,s\,; N;
\nonumber  \\
 \begin{cases} \hat\gamma I_9>0, \hat\beta\hat\gamma>0,
   I_9(4-\hat\gamma)\le0,   \\ \text{or }
   \hat\gamma=0,\; \hat\beta<0  \end{cases}
      \Leftrightarrow    \emph{Vul}_2 \Rightarrow
\begin{cases} c,\is\,; N
\text{ if } \hat\gamma(\hat\gamma-4)(\hat\gamma-6)\ne0;\\
               c, \is\,; \overline{2\choose1}  {N}
\text{ if }  \hat\gamma=0;\\
               c, \is\,; \overline{0\choose3}  {N}
\text{ if }  \hat\gamma=4;\\
               c,\is\,; N
\text{ if }  \hat\gamma=6;\\
\end{cases}
\nonumber  \\
I_9<0,\;   0\le\hat\gamma\le4,\; \hat\beta>0     \Leftrightarrow
 \emph{Vul}_{20} \Rightarrow
\begin{cases} c, c\,;  S
 \text{ if }\hat\gamma(\hat\gamma-4)\ne0;\\
           c, c\,; \overline{2\choose1}  {S}
\text{ if } \hat\gamma=0;\\
           c,c\,; \overline{0\choose3}  {S}
\text{ if } \hat\gamma=4;
 \end{cases}
\nonumber  \\
I_9<0,\;  \hat\gamma>4,\; \hat\beta>0    \Leftrightarrow
 \emph{Vul}_{21} \Rightarrow  c,c\,; S,S, N;
  \\
 I_9>0,\;   0\le\hat\gamma<4     \Leftrightarrow
 \emph{Vul}_{19} \Rightarrow
\begin{cases} c, \is\,;  S,N,N
\text{ if }  \hat\gamma \ne0;\\
              c, \is\,; \overline{2\choose1}  {S},N,N
\text{ if } \hat\gamma=0;\\
\end{cases}
\nonumber \\
 \hat\beta<0,\;   0<\hat\gamma\le4     \Leftrightarrow
 \emph{Vul}_{25}  \Rightarrow
\begin{cases} c, \is,n,n\,;  S
\text{ if }  \hat\gamma \ne4;\\
             c, \is,n,n\,; \overline{0\choose3}  {S}
\text{ if }  \hat\gamma=4;\\
 \end{cases} \nonumber
 \end{gather}
and
\begin{equation}\label{cond:ph-port-S3-1B}
\begin{split}
 &(i) \quad \hat\gamma I_9<0,\;  \hat\gamma( \hat\gamma-6)>0
    \Leftrightarrow  \emph{Vul}_8,\emph{Vul}_9,\emph{Vul}_{10}
\Rightarrow  c,\is,s,s\,; N,N, N;\\
 &(ii) \quad I_9<0,\;   \hat\gamma=6    \Leftrightarrow
  \emph{Vul}_{16},\emph{Vul}_{17},\emph{Vul}_{18}
  \Rightarrow  c, \is\,;\, N ,\overline{1\choose 1}{ SN },   \overline{1\choose 1}
   { SN };\\
 &(iii) \quad \hat\beta<0,\;   4<\hat\gamma<6
    \Leftrightarrow  \emph{Vul}_{26},\emph{Vul}_{27},\emph{Vul}_{28}
 \Rightarrow  c,\is,n,n\,; S,S, N.\\
 \end{split}
\end{equation}

On the other hand for  systems $(S^{(c)}_3)$ we calculate
\begin{equation}\label{val:S3-AFF}
\begin{gathered}
\mu_0=4de(e-1)^2,\quad \mathbf{D}=192e(d+2-2e)^3,\quad
\eta=4d(2-3e)^3,\\
\widetilde K=4(e-1)(dx^2-ey^2),\quad \mu_1=4e(e-1)(d+e-1)y,\\
\mu_2\big|_{e=0}=d(d+2)x^2,\quad
 \mu_2\big|_{e=1}=d(d x^2+y^2).
 \end{gathered}
\end{equation}
So considering \eqref{val:S3-GL} and \eqref{val:S3-AFF}  if
$\mathbf{D}\mu_0\eta\ne0$ we obtain
$$
\operatorname{sign}(\mu_0) =\operatorname{sign}( \hat\gamma I_9),\quad \operatorname{sign}(\mathbf{D})=
\operatorname{sign}(\hat\beta\hat\gamma),\quad
\operatorname{sign}(\eta)=\operatorname{sign}(I_9(4-\hat\gamma)),
$$
and due to $\mathcal{T}_2\ne0$ we have that $\mu_0=0$ (respectively
$\mathbf{D}=0$; $\eta=0$) if and only if
$\hat\gamma(\hat\gamma-6)=0$ (respectively $\hat\gamma=0$;
$\hat\gamma=4$). Therefore it is not too hard to determine that in
cases $(6.9)$ (when we have the unique phase portrait) the
conditions from Table \ref{table2} (the case of statement (c) of
Proposition \ref{prop:weak-sing}) are equivalent to the respective
conditions from $(6.9)$.

We consider now the remaining cases \eqref{cond:ph-port-S3-1B}.
According to \cite{Vulpe-DU} the phase portraits
\emph{Vul}$_{\,8},$ $\emph{Vul}_9$, $\emph{Vul}_{10}$
(respectively \emph{Vul}$_{\,16},\emph{Vul}_{17},\emph{Vul}_{18}$;
\emph{Vul}$_{\,26},\emph{Vul}_{27},\emph{Vul}_{28}$) are
distinguished via the $GL$-invariant $\hat\gamma I_3I_4$.
More precisely in the mentioned cases the phase portrait
corresponds to \emph{Vul}$_{\,8}$ (respectively \emph{Vul}$_{\,16}$;
\emph{Vul}$_{\,26}$)
if $\hat\gamma I_3I_4>0$; \emph{Vul}$_{\,9}$ (respectively
\emph{Vul}$_{\,18}$; \emph{Vul}$_{\,28}$) if $\hat\gamma I_3I_4<0$
and it corresponds to \emph{Vul}$_{\,10}$ (respectively
\emph{Vul}$_{\,17}$; \emph{Vul}$_{\,27}$) if $I_3=0$.

On the other hand for systems $(S^{(c)}_3)$ we calculate
$$
B_3 =-6 (2 + d - 3 e) (d + e) x^3 y,\quad B_3B_5= 288 d (2 + d - 3
e) e (d + e) x^4 y^2.
$$
We claim that in all three cases \eqref{cond:ph-port-S3-1B}, if
$B_3\ne0$ then we have \begin{equation}\label{sign_B3B5}
\operatorname{sign}(B_3B_5)=\operatorname{sign}(\hat\gamma I_3I_4)=\operatorname{sign}\big(e(d+e)\big), \end{equation}
and $B_3=0$ if and only if $I_3=0$ (i.e. $d+e=0$). To prove this
claim we shall consider each one of the cases  $(i)-(iii)$ from
\eqref{cond:ph-port-S3-1B}.

\emph{Case}  (i). Considering \eqref{val:S3-GL} we have $de<0$,
$e(e-1)>0$ and herein it  can easily be detected that
$\operatorname{sign}(d+2-3e)=-\operatorname{sign}(e)$ and this leads to  \eqref{sign_B3B5}.

\emph{Case}  (ii). As $e=1$ we have $B_3B_5=288 d(d-1)(1 + d)
x^4 y^2$ and due to   $d<0$ this evidently implies
\eqref{sign_B3B5}.

\emph{Case} (iii). In this case considering \eqref{val:S3-GL}
we have $2/3<e<1$ and   $d+2-2e<0$. Therefore $d<0$ and $2-3e<0$
that gives $ d+2-3e<0$. As $e>0$ we again obtain \eqref{sign_B3B5}.

It remains to note that due to the conditions discussed above in all
three cases we can have $B_3=0$ if and only if $d+e=0$ (i.e.
$I_3=0$).

Thus our claim is proved and obviously we arrive to the conditions of
Table \ref{table3} corresponding to the configurations 8, 9 and 22
respectively.


\subsubsection\textbf{The case $\mathcal{T}_2=0$}

Then $d(d+2-2e)=0$ and considering \eqref{val:S3-GL}  for systems
$(S^{(c)}_3)$ we have $I_9\hat\beta=0$. We recall that by
Proposition \ref{prop:weak-sing} in this case for a system
$(S^{(c)}_3)$ has to be satisfied  the conditions of the statement
$(e)$, i.e. besides the center we could not have another weak
singularity. So according to \cite{Vulpe-DU}
(see the proof of Lemma 3) the phase portrait
(and hence, the respective configuration of
singularities) of a system from the family $(S^{(c)}_3)$ with the
condition $I_9\hat\beta=0$ is determined by the following
$GL$-invariant conditions
\begin{equation}\label{cond:ph-port-S3-2}
\begin{gathered}
  I_9=0,\; \hat\gamma(\hat\gamma-6)>0  \Leftrightarrow
 \emph{Vul}_5 \Rightarrow  c,s,s\,;
  N,\widehat{1\choose 2} P\;E\;P - H;
\\
   I_9=0,\;   0\le\hat\gamma\le3   \Leftrightarrow \emph{Vul}_{12}
   \Rightarrow   \begin{cases}
c\,;  S, \widehat{1\choose 2} P\;E\;P - H &\text{if } \hat\gamma\ne0;\\
c\,; \overline{ {2\choose 1}  }\ {S},\widehat{1\choose 2} P\;E\;P -  H
&\text{if } \hat\gamma=0;
    \end{cases}
 \\
I_9=0,\;   3<\hat\gamma<4   \Leftrightarrow \emph{Vul}_{14}
\Rightarrow  c\,; S, \widehat{1\choose 2} P\;E\;P -  P\;H\;P;\\
I_9=0,\;    \hat\gamma=4    \Leftrightarrow  \emph{Vul}_{15}
\Rightarrow  c\,; [\infty,S];
\\
I_9=0,\;   4<\hat\gamma\le6    \Leftrightarrow \emph{Vul}_{13}
 \Rightarrow   \begin{cases} c\,;   N, \widehat{1\choose2}H-HHH & \text{if }
   \hat\gamma\ne6;\\
    c\,;   N, \widehat {3\choose2}H-HHH &\text{if } \hat\gamma=6;
    \end{cases}
\end{gathered}
\end{equation}
and
  \begin{equation}\label{cond:ph-port-S3-2a}
\begin{gathered}
 \hat\beta=0,\;   \hat\gamma I_9>0,  I_9(4-\hat\gamma)<0
\Leftrightarrow  \emph{Vul}_2 \Rightarrow    c,
 \overline{s}_{(3)}\,; {N};
\\
   \hat\beta=0,\;   0<\hat\gamma<3   \Leftrightarrow
 \emph{Vul}_{22}   \Rightarrow   c, \widehat{es}_{(3)}\,;  S;
\\
   \hat\beta=0,\; 3\le\hat\gamma\le4    \Leftrightarrow
 \emph{Vul}_{23} \Rightarrow \begin{cases}
c, \widehat{es}_{(3)}\,;  S &\text{if } \hat\gamma\ne4;\\
c, \widehat{es}_{(3)}\,; \overline{ {0\choose 3} } S &\text{ if }\hat\gamma=4;\\
 \end{cases}
\\
   \hat\beta=0,\;   4<\hat\gamma<6   \Leftrightarrow  \emph{Vul}_{24}
  \Rightarrow   c, \widehat{es}_{(3)}\,;  S, S, N;
\\
  \hat\beta=0,\;  \hat\gamma=0  \Leftrightarrow  \emph{Vul}_{29}
  \Rightarrow   c, \big(  \boldsymbol{\ominus}[|];\ses \big);\
   \big(  \boldsymbol{\ominus}[|];\emptyset\big).
\end{gathered}
\end{equation}
We remark that by \eqref{val:S3-GL}  the condition $I_9=0$
gives $d=0$ whereas the condition  $\hat\beta=0$   gives $d=2(e-1)$.

\textbf{(1)} Assume first $I_9=0$, i.e. $d=0$. Then for systems
$(S^{(c)}_3)$ we obtain
\begin{equation}\label{val:S3-AFF-1a}
\begin{gathered}
\mu_0=\eta=0,\quad \mathbf{D}=-1536e (e-1)^3, \quad
\widetilde N=4(1-e)(2e-1)y^2,\\
\tilde L=8e (3e-2)y^2,\quad \mu_1=4e(e-1)^2y,\quad
 \mu_3 = 2 ( 1 - e) x^2 y + e y^3.
\end{gathered}
\end{equation}
As  $\hat\gamma=6e$ (see \eqref{val:S3-GL})
this implies $\hat\gamma-6=6(e-1)$, $\hat\gamma-4=2(3e-2)$ and
$\hat\gamma-3=3(2e-1)$. So if $\hat\gamma(
\hat\gamma-3)(\hat\gamma-4)(\hat\gamma-6)\ne0$ then we have
\begin{equation}\label{signs_(gamma-j)}
 \begin{gathered}
\operatorname{sign}(\mathbf{D})=-\operatorname{sign}\big(\hat\gamma
(\hat\gamma-6)\big),\quad
\operatorname{sign}(\widetilde N)\big|_{\{\mathbf{D}>0\}}
= \operatorname{sign}(\hat\gamma-3\big),\\
\operatorname{sign}(\tilde L)\big|_{\{\mathbf{D}>0,\widetilde N>0\}}=
\operatorname{sign}(\hat\gamma-4\big).
\end{gathered} \end{equation}
Moreover if $\mu_1\ne0$ then
$\widetilde N =0$  (respectively $\tilde L =0$) if and only if
$\hat\gamma=3$ (respectively $\hat\gamma=4$). Therefore to determine
the cases \eqref{cond:ph-port-S3-2} (when $I_9=0$)   the conditions
of Table \ref{table2} (the case of statement (e) of Proposition
\ref{prop:weak-sing}) are equivalent to the respective conditions of
\eqref{cond:ph-port-S3-2}.
\smallskip

\textbf{(2)} Suppose now $\hat\beta=0$ and $I_9\ne0$. Then
$d=2(e-1)\ne0$ and for systems $(S^{(c)}_3)$ we have
\begin{equation}\label{val:S3-AFF-1b}
\begin{gathered}
\mu_0=8e(e-1)^3,\quad \eta=8(e-1)(2-3e)^3,\quad
 \mu_1=12e(e-1)^2y, \\
\mu_3=ey^3, \quad  \theta_1=128 (e-1) (2e-1) (3e-2) (5 - 6 e).
\end{gathered}
\end{equation}
We observe that if $\mu_0\ne0$ then
\begin{equation}\label{signs_(gamma-j)-B}
\begin{gathered}
\operatorname{sign}(\mu_0)= \operatorname{sign}\big(\hat\gamma
(\hat\gamma-6)\big),\quad
\operatorname{sign}(\eta)\big|_{\{\mu_0<0\}}
= \operatorname{sign}(\hat\gamma-4\big),\\
\operatorname{sign}(\theta_1)\big|_{\{\mu_0<0,\eta<0\}}
= \operatorname{sign}(\hat\gamma-3\big).
\end{gathered}
\end{equation}
We claim that the conditions $\hat\gamma I_9>0 $ and
$I_9(4-\hat\gamma)<0$ corresponding to the phase portrait $\emph{
Vul}_2$ (see \eqref{cond:ph-port-S3-2a}) are equivalent to
$\mu_0>0$. Indeed, as $I_9=d=2(e-1)\ne0$ we obtain $\hat\gamma
I_9=12e(e-1)$ and hence the condition $\hat\gamma I_9>0 $ is
equivalent to $\mu_0>0$. It remains to note that in the case
$e(e-1)>0$ we have $I_9(4-\hat\gamma)=-4(e-1)(3e-2)<0$. So
$3e-2\ne0$ (i.e. $\hat\gamma\ne4$) and we arrive to the respective
conditions from Table \ref{table2}.

Considering  the remaining cases \eqref{cond:ph-port-S3-2a}
corresponding to the condition $\hat\beta=0$ and
\eqref{signs_(gamma-j)-B} we arrive to the respective  conditions
provided by Table \ref{table2} (the case of statement
(e) of Proposition
\ref{prop:weak-sing}).

\subsection{The family of  systems $(S^{(c)}_4)$}

For these systems we have $\sigma=0$; i.e., this is a class of
Hamiltonian  systems with center. Moreover any elemental point which
is not a center must be an integrable saddle. Calculations yield
\begin{equation}\label{val:S4-GL}
\begin{gathered}
\hat\alpha= 8 (a - 2 c)^2 (b^2 - 4 a c + 8 c^2),\quad I_8=2 a (a
b^2 + 4 c^3),\\
I_{10} =-b^2 - (a + c)^2,\quad I_{16}=b(3 a c^2-a^3 + a b^2  + 2 c^3).
\end{gathered}
\end{equation}
According to \cite{Vulpe-DU} (see the proof of Lemma 2) the
phase portrait (and hence, the respective configuration of
singularities) of a quadratic system from the family $(S^{(c)}_4)$ is
determined by the following $GL$-invariant conditions
\begin{equation}\label{cond:ph-port-S4-1}
\begin{gathered}
\begin{cases}
\hat\alpha<0,\text{ or}\\
\hat\alpha=I_{16}= 0 \end{cases}
 \Leftrightarrow  \emph{Vul}_2
\Rightarrow \begin{cases} c, \is\,;    N
&\text{if } \hat\alpha\ne0;\\
 c,   \overline{s}_{(3)};  {N}
&\text{ if }  \hat\alpha=0,\; I_8\ne0;\\
 c, \is\,;  \widehat{2\choose 3} N
&\text{if } \hat\alpha=0,\; I_8=0;
 \end{cases}
\\
\hat\alpha>0,\;  I_8<0    \Leftrightarrow
\emph{Vul}_8,\emph{Vul}_9,\emph{Vul}_{10}, \emph{Vul}_{11}
\Rightarrow c, \is ,\is ,\is\,;   N, N, N;\\
\hat\alpha>0,\;  I_8>0    \Leftrightarrow
\emph{Vul}_3,\emph{Vul}_4 \Rightarrow  c, c ,\is ,\is\,;  N;\\
\hat\alpha>0,\;  I_8=0    \Leftrightarrow \emph{Vul}_5,\emph{Vul}_6
\Rightarrow  c, \is ,\is\,;  N,
\widehat{1\choose 2}PEP-H;\\
\hat\alpha=0,\;  I_{16}\ne0   \Leftrightarrow \emph{Vul}_7
\Rightarrow  c,\is , \widehat{cp}_{(2)}\,;  N.
\end{gathered}
\end{equation}
On the other hand for systems $(S^{(c)}_4)$ calculations
yield
\begin{equation}\label{val:S4-Aff}
\begin{gathered} \mu_0=a (a b^2 + 4 c^3)=
I_8/2,\quad \mathbf{D}=-48 (a - 2 c)^2 (b^2 - 4 a c + 8
c^2)=-6\hat\alpha,\\ \mu_1=2 a b (a - c) x - 2 (a b^2 + 2 a c^2 +
2 c^3) y.
\end{gathered}
\end{equation}
We observe  that the condition $I_8<0$
implies $c\ne0$ (then  by Proposition \ref{prop:GL-can-syst} we
have $c=1/2$) and $a<0$. In this case   we evidently obtain
$\hat\alpha>0$. Similarly  the condition $\hat\alpha<0$ gives
$c\ne0$ (i.e. $c=1/2$) and $a>1+b^2/2$ and this implies $I_8>0$.
Herein we conclude that in the case $I_8\hat\alpha\ne0$ (i.e.
$\mu_0\mathbf{D}\ne0$) the conditions provided by Table \ref{table2} (the case of
statement (f) of Proposition \ref{prop:weak-sing}) for
distinguishing the configurations of the singularities are
equivalent to the respective conditions of
\eqref{cond:ph-port-S4-1}.

Assume now $I_8\hat\alpha=0$ (i.e. $\mu_0\mathbf{D}=0$).
\smallskip

\textbf{(1)} If  $I_8=0$ then by \eqref{val:S4-GL} we have $a (a
b^2 + 4 c^3)=0$ and then $\mu_0=0$.  We claim that in the case
$\hat\alpha\ne0$ we obtain  $\hat\alpha>0$ and this is equivalent
to $\mu_1\ne0$.  Indeed as the condition $\hat\alpha\ne0$ implies
$(a^2+c^2)(b^2+c^2)\ne0$, we conclude that the condition $I_8=0$
gives  $a\le0$ and $c\ne0$. So  $c=1/2$ and we obtain
$\hat\alpha>0$. On the other hand since $a-c\ne0$ we obtain that
$\mu_1=0$ if and only if $ab=a+c=0$ but in this case the condition
$I_8=0$ implies $\hat\alpha=0$. So our claim is proved and this
shows the equivalence of the respective conditions of Table \ref{table2} and
\eqref{cond:ph-port-S4-1}.

Assume $I_8=0=\hat\alpha$. Then considering
\eqref{cond:ph-port-S4-1} we obtain portrait $\emph{Vul}_2$ and the
configuration of singularities indicated in the row 41 of Table \ref{table2}.
It remains to observe that the condition above is equivalent to
$\mu_0=\mu_1=0$.

\textbf{(2)} Suppose now $ \hat\alpha=0$ and $I_8\ne0$. This implies
$\mu_0\ne0$ and $\mathbf{D}=0$, i.e. $(a - 2 c) (b^2 - 4 a c + 8 c^2)$=0. So
we obtain
$$
\mathbf{T}= -48 b^2 c^4 y^2 (2 c x + b y)^2 (b x - c y)^2,\ \ I_{16}=2 b^3
c,\ \ \mu_0=4 c^2 (b^2 + 2 c^2),
$$
if $a=2c$ and
$$
\begin{gathered} \mathbf{T}= - \frac{3b^2 }{ 4096 c^6 }\, (b^2 + 4 c^2)^4 y^2 (b x - 2 c
y)^2 (b^2 x + 8 c^2 x + 2 b c y)^2,\\ I_{16}=-\frac{1}{64 c^3}\, b^3
(b^2 + 4 c^2)^2,\ \ \mu_0=\frac{1}{16 c^2}(b^2 + 4 c^2)^2 (b^2 + 8
c^2), \end{gathered}
$$
if $a=(b^2+8c^2)/(4c)$ (we note that $c\ne0$ due to $I_8\ne0$).

Thus clearly if $ \hat\alpha=0$ and $I_8\ne0$ then the condition
$I_{16}=0$ is equivalent to $\mathbf{T}=0$ and as $\mu_0>0$ we obtain that
the respective  conditions provided by Table \ref{table2}
(see rows 38 and \ref{table3}9)
are equivalent to the corresponding conditions from
\eqref{cond:ph-port-S4-1}.

To finish the proof of Theorem 1 it remains to examine the
conditions for distinguishing  the different phase portraits which
correspond to the same configuration of singularities. We have
three groups of such phase portraits: $(i)\ \emph{Vul}_8-\emph{
Vul}_{11}$; $(ii)\ \emph{Vul}_3,\emph{Vul}_4$ and $(iii)\ \emph{
Vul}_5,\emph{Vul}_6$. According to \cite{Vlp_prepr_B-2010}
quadratic systems $(S^{(c)}_4)$ possess one of the mentioned phase
portraits if and only if the following conditions are fulfilled
\begin{equation}\label{cond:ph-port-S4-2}
 \begin{gathered}
   \hat\alpha>0,\;  I_8>0    \Rightarrow    \begin{cases}
    \emph{Vul}_3 & \text{if }   I_{16}=0;\\
    \emph{Vul}_4 & \text{if } I_{16}\ne0;\\
   \end{cases}
\\
   \hat\alpha>0,\;  I_8=0    \Rightarrow   \begin{cases}
    \emph{Vul}_5 & \text{if }  I_{16}=0;\\
    \emph{Vul}_6 & \text{if } I_{16}\ne0;\\
   \end{cases}
\\
   \hat\alpha>0,\;  I_8<0    \Rightarrow    \begin{cases}
    \emph{Vul}_8 & \text{if } I_{10}\ne0,  I_{16}=0,  \hat\xi>0;\\
    \emph{Vul}_9 & \text{if } I_{10}\ne0,  I_{16}=0,  \hat\xi<0;\\
    \emph{Vul}_{10} & \text{if } I_{10}=0;\\
    \emph{Vul}_{11} & \text{if }   I_{16}\ne0.
   \end{cases}
\\
\end{gathered}
 \end{equation}
On the other hand for systems $(S^{(c)}_4)$ we have
$$
\begin{gathered}
 B_1 =-I_{16} [b^2+(a-3c)^2],\quad  I_{16}=b(3 a c^2-a^3 + a
b^2  +
2 c^3),\\
B_3= -3 a b x^4 - 6 (a - 3 c) (a + c) x^3 y + 18 b c x^2 y^2 +
 6 b^2 x y^3 + 3 b (a - 2 c) y^4,
 \end{gathered}
$$
and hence the condition $I_{16}=0$ is equivalent to $B_1=0$.
Herein we arrive to the conditions provided by  Table \ref{table3} for the
portraits $\emph{Vul}_3-\emph{Vul}_6$ respectively.

Next we examine the conditions for the phase portraits $\emph{
Vul}_8-\emph{Vul}_{11}$. First we observe that by \eqref{val:S4-GL}
the condition $I_{10}=0$ yields $b=a+c=0$ and this implies
$I_{16}=B_3=B_1=0$. Therefore in the case $I_{16}\ne0$ (this is
equivalent with $B_1\ne0$)  we get \emph{Vul}$_{\,11}$ and in the
case $I_{10}=0$ (then $B_3=0$) we obtain \emph{Vul}$_{\,10}$ (see
Table \ref{table3}, configuration number 35). So it remains to consider the
phase portraits \emph{Vul}$_{\,8}$ and \emph{Vul}$_{\,9}$. We claim
that in the case $I_{16}=0$ we have
$\operatorname{sign}(\hat\xi)=\operatorname{sign}(B_3B_4)$. Indeed assuming $I_{16}=0$ we
shall examine two cases, $b=0$ and $b\ne0$.
\smallskip

\textbf{(1)} If $b=0$ (then $I_{16}=0$) a straightforward calculation
for systems $(S^{(c)}_4)$ yields
$$
\hat\xi=-4 (a - c) (a + c)^3,\ \ B_3B_4=-192 (a - 3 c)^3 c^4 (a +
c)^3 x^4 y^2,\ \ I_8=8ac^3,
$$
and as $I_8<0$ (i.e. $ac<0$ ) we get $(a-c)(a-3c)>0$. So clearly our
claim is proved in this case.
\smallskip

\textbf{(2)} Assume now  $b\ne0$.  Then $I_8<0$ gives $ac<0$ and hence
we can set a new parameter $u$ as follows $u^2=(a-2c)/a$. Herein we
have $c=a(1-u^2)/2$ and   calculation gives
$$
I_{16}=ab(2 b - 3 a u + a u^3)(2 b + 3 a u - a u^3)/4.
$$
Hence  due to $ab\ne0$ the  condition $I_{16}=0$ gives $b=\pm
au(u^2-3)/2$ and then we calculate
$$
\begin{gathered} \hat\xi= -a^4 (u^2-1) (u^2-3)^3 (1 + u^2)^2/2,\
\ I_8= a^4 (2 - u^2) (1 + u^2)^2/4,\\
B_3B_4=-3a^{10} (u^2-3)^3 (1 + u^2)^5 (u x \pm y)^2 ( u y\mp
x)^4/16. \end{gathered}
$$
As $I_8<0$ we have $u^2-2>0$ and this implies $u^2-1>0$. Therefore
$\operatorname{sign}(\hat\xi)=\operatorname{sign}(B_3B_4)$, i.e. our claim is valid and we
arrive to the respective  conditions given by Table \ref{table3} in the
considered case.

As all the cases are examined Theorem \ref{thm:QS-center} is proved.
\hfill $\Box$


\section{The family of Hamiltonian quadratic systems}\label{sec:hamil}


In this section  to prove Theorem  \ref{thm:QS-Hamilt} we
need   the following invariant polynomials defined in
\cite{KalVul} using the invariant polynomials
\eqref{inv-com:tens-form} and \eqref{inv-com:tens-form-1} (we keep
the respective notations adding only the ``hat'')
\begin{equation}\label{comit:Ham-Sys} \begin{gathered}
 2\widehat \mu= I_8,\quad \widehat H= -K_{14},\quad
4\widehat G= -5I_2K_7+2I_5K_2+4K_3^2+8K_{31},
\\
2\widehat F= -I_2K_{11}-4I_{19}K_5+4K_2K_{27},\quad
\widehat V= K_{11}K_{22}+K_{23}^2,
\\
\widehat R = 3\widehat H^2-2\widehat G\widehat \mu, \quad
\widehat S = 2\widehat F\widehat H{\widehat \mu}^2+
         \widehat G^2{\widehat \mu}^2-4\widehat G\widehat H^2\widehat \mu
         +3\widehat H^4-4{\widehat \mu}^3\widehat V,
\\
\widehat P = \widehat G^2-6\widehat F\widehat H+12\widehat \mu
 \widehat V, \quad
\widehat U = \widehat F^2-4\widehat G\widehat V,
\\
\widehat T = 9\widehat F^2{\widehat \mu}^2-14\widehat
 F\widehat G\widehat H\widehat \mu+12\widehat F\widehat H^3
 +2\widehat G^3\widehat \mu-2\widehat G^2\widehat H^2
 -8\widehat G{\widehat \mu}^2\widehat V+12\widehat H^2\widehat \mu \widehat V,
\\
\widehat W_1 = K_2^2-4K_5K_{21}, \quad \widehat  W_2 =
-K_2K_{12}-2K_3K_{11}+4K_5K_{27}+6K_7K_{23},
\\
\widehat E_1 = 4I_{10}-5I_2I_5-24I_{30}, \quad
 \widehat  E_2 = I_2^2+32I_{23}-16I_{24},
\\
\widehat E_3 = I_2^3+32(12I_5I_{18}-3I_{19}^2+16I_{33}),
\\
32\widehat D = 4I_5^2(3I_2\widehat E_2-\widehat E_3)-\widehat
E_2(4I_5\widehat E_1+I_8\widehat E_2),
\\
\widehat T_c = (9I_5^3+I_8\widehat E_1)^2+9I_5^2(I_8^2\widehat
E_2-3I_5^4) +I_8^3(3I_2\widehat E_2-\widehat E_3),
\\
\widehat U_c = 3I_8I_{19}(3I_2I_5+16I_{30})
-2I_8(3I_2I_{35}+24I_5I_{28}+16I_8I_{21})
\\
+3I_5^2(I_5I_{19}+10I_{35}) -2I_{16}(2I_2I_5-I_{10}+12I_{30}).
\end{gathered}
\end{equation}

\begin{table}[!ht]\scriptsize
\caption{} \label{table4}
\begin{center}
\begin{tabular}{|@{}c@{}|@{}c@{} |@{} c @{}|@{} c @{}|@{} c @{}|@{} c @{}| c |@{}c@{}|}
 \hhline{|=======} \multicolumn{5}{|c|}{\begin{tabular}{c}\emph{Affine invariant  conditions for configurations} \end{tabular}} &
 \begin{tabular}{c}\emph{Configuration}\\[-1mm] \emph{of singularities}\end{tabular} & No.   \rule{0pt}{5.0mm}\\[0.3mm]
 \hline
   \multicolumn{1}{|@{}c }{\multirow{6}*{\raisebox{-0.2em}[0pt][0pt]{ $\begin{array}{c}\mu_0<0\end{array}$}}}
  &\multicolumn{4}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\mathbf{D}<0\end{array}$}}}
     & $\is ,\is ,\is ,c\,;\  N, N, N$ & 1\rule{0pt}{3.5mm}\\[0.3mm]
 \cline{2-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{4}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\mathbf{D}>0\end{array}$}}}
     & $\is ,\is \,;\ N, N, N $ &2\rule{0pt}{3.5mm}\\[0.3mm]
\cline{2-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{\multirow{3}*{ \raisebox{-0.2em}[0pt][0pt]{$\begin{array}{c}\mathbf{D}=0\end{array}$}}}
  &\multicolumn{3}{|c|}{$  \begin{array}{c}\mathbf{T}\ne0\end{array}  $}
     & $\is , \is ,\widehat{cp}_{(2)}\,;\ N, N, N$ &3\rule{0pt}{3.5mm}\\[0.3mm]
\cline{3-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{\raisebox{-0.7em}[0pt][0pt]{$  \begin{array}{c}\mathbf{T}=0\end{array}  $}}
   &\multicolumn{2}{|c|}{$  \begin{array}{c}\mathbf{R}\ne0\end{array}  $}
     & $\is , \widehat s_{(3)}\,;\ N,N,N $ &4\rule{0pt}{3.5mm}\\[0.3mm]
\cline{4-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{ }
   &\multicolumn{2}{|c|}{$  \begin{array}{c}\mathbf{R}=0\end{array}  $}
     &\,$(hhhhhh)_{(4)}\,;\, N,N,N $\, &5\rule{0pt}{3.5mm}\\[0.3mm]
 \hline
   \multicolumn{1}{|@{}c }{\multirow{9}*{\raisebox{-0.3em}[0pt][0pt]{ $\begin{array}{c}\mu_0>0\end{array}$}}}
  &\multicolumn{1}{|c|}{\multirow{1}*{ \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}\mathbf{D}<0\end{array}$}}}
    &\multicolumn{3}{|c|}{$(\mathbf{R}>0)\&(\mathbf{S}>0)$}
     & $\is ,\is ,c ,c\,;\   N$ &6\rule{0pt}{3.5mm}\\[0.3mm]
\cline{3-7}
   \multicolumn{1}{|@{}c }{ }
  &\multicolumn{1}{|c|}{ }
    &\multicolumn{3}{|c|}{$(\mathbf{R}\le0)\vee(\mathbf{S}\le0)$}
     & $ \ses ; N$ &7\rule{0pt}{3.5mm}\\[0.3mm]
\cline{2-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{4}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\mathbf{D}>0\end{array}$}}}
     & $\is , c\,;\ N$ &8\rule{0pt}{3.5mm}\\[0.3mm]
\cline{2-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{\multirow{6}*{ \raisebox{-0.3em}[0pt][0pt]{$\begin{array}{c}\mathbf{D}=0\end{array}$}}}
  &\multicolumn{3}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\mathbf{T}<0\end{array}$}}}
     & $\is , c,\widehat{cp}_{(2)}\,;\  N  $ &9\rule{0pt}{3.5mm}\\[0.3mm]
\cline{3-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{3}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\mathbf{T}>0\end{array}$}}}
     & $ \widehat{cp}_{(2)}\,;\  N  $ &10\rule{0pt}{3.5mm}\\[0.3mm]
\cline{3-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{\multirow{4}*{ \raisebox{-0.3em}[0pt][0pt]{$\begin{array}{c}\mathbf{T}=0\end{array}$}}}
  &\multicolumn{2}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\mathbf{P}\mathbf{R}<0\end{array}$}}}
     & $   \ses ; N  $ &11\rule{0pt}{3.5mm}\\[0.3mm]
\cline{4-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{2}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\mathbf{P}\mathbf{R}>0\end{array}$}}}
     & $   \widehat{cp}_{(2)},\widehat{cp}_{(2)}\,;\, N  $ &12\rule{0pt}{3.5mm}\\[0.3mm]
\cline{4-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{\multirow{1}*{ \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}\mathbf{P}\mathbf{R}=0\end{array}$}}}
  &\multicolumn{1}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c} \mathbf{R}\ne0\end{array}$}}}
     & $  c, \widehat{s}_{(3)}\,;\, N  $ &13\rule{0pt}{3.5mm}\\[0.3mm]
 \cline{5-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{\multirow{1}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c} \mathbf{R}=0\end{array}$}}}
     & $  (hh)_{(4)}\,;\, N  $ &14\rule{0pt}{3.5mm}\\[0.3mm]
    %%%%%%%%%
    %%%%%%%%%
    %%%%%
 \hline
   \multicolumn{1}{|@{}c }{\multirow{17}*{\raisebox{-0.0em}[0pt][0pt]{ $\begin{array}{c}\mu_0=0\end{array}$}}}
  &\multicolumn{1}{|c|}{\multirow{4}*{ \raisebox{-0.2em}[0pt][0pt]{$\begin{array}{c}\mu_1\ne0\end{array}$}}}
  &\multicolumn{3}{|c|}{$  \begin{array}{c}\mathbf{D}<0\end{array}  $}
     & $\is ,\is , c\,;\ N, \widehat{1\choose 2}PEP-H$  &15\rule{0pt}{5.0mm}\\[0.3mm]
\cline{3-7}
   \multicolumn{1}{|@{}c }{}
   &\multicolumn{1}{|c|}{}
  &\multicolumn{3}{|c|}{$  \begin{array}{c}\mathbf{D}>0\end{array}  $}
     & $\is\,;\  N, \widehat  {1\choose 2}\, PEP-H$& 16\rule{0pt}{5.0mm}\\[0.3mm]
\cline{3-7}
   \multicolumn{1}{|@{}c }{}
   &\multicolumn{1}{|c|}{}
  &\multicolumn{1}{|c|}{$  \begin{array}{c}\mathbf{D}=0\end{array}  $}
  &\multicolumn{2}{|c|}{$  \begin{array}{c}\mathbf{P}\ne0\end{array}  $}
     & \,$\is ,\widehat{cp}_{(2)}\,;\  N, \widehat{1\choose 2}\,P\;E\;P-H$ & 17\rule{0pt}{5.0mm}\\[0.3mm]
\cline{4-7}
   \multicolumn{1}{|@{}c }{}
   &\multicolumn{1}{|c|}{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{2}{|c|}{$  \begin{array}{c}\mathbf{P}=0\end{array}  $}
     & \,$\widehat{s}_{(3)}\,;\  N, \widehat{1\choose 2}\, PEP-H$&   18\rule{0pt}{5.0mm}\\[0.3mm]
\cline{2-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{\multirow{11}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\mu_1=0\end{array}$}}}
  &\multicolumn{1}{|c|}{\multirow{6}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\mu_2\ne0\end{array}$}}}
  &\multicolumn{1}{|c|}{\raisebox{-0.7em}[0pt][0pt]{$  \begin{array}{c}\mathbf{U}<0\end{array}  $}}
  &\multicolumn{1}{|c|}{$  \begin{array}{c}\widetilde M\ne0\end{array}  $}
    & $   \ses ;  N, {2\choose 2} H-H$& 19\rule{0pt}{4.0mm}\\[0.3mm]
\cline{5-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{ }
 &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{$  \begin{array}{c}\widetilde M=0\end{array}  $}
   & $ \ses ; \widehat{2\choose 3}\,  N $& 20\rule{0pt}{5.0mm}\\[0.3mm]
\cline{4-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{\raisebox{-0.7em}[0pt][0pt]{$  \begin{array}{c}\mathbf{U}>0\end{array}  $}}
  &\multicolumn{1}{|c|}{$  \begin{array}{c}\widetilde M\ne0\end{array}  $}
    & $\is ,\is \,;\,   N, {2\choose 2} P\;E\;P-P\;E\;P$& 21\rule{0pt}{4.0mm}\\[0.3mm]
\cline{5-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{ }
 &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{$  \begin{array}{c}\widetilde M=0\end{array}  $}
   & $\is , c\,;\ \widehat{2\choose 3}\,  N $& 22\rule{0pt}{5.0mm}\\[0.3mm]
\cline{4-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{ }
   &\multicolumn{2}{|c|}{$  \begin{array}{c}\mathbf{U}=0\end{array}  $}
   & $\widehat{cp}_{(2)}\,;\ \widehat{2\choose 3}\,  N $& 23\rule{0pt}{5.0mm}\\[0.3mm]
\cline{3-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{\multirow{5}*{ \raisebox{-0.0em}[0pt][0pt]{ $\begin{array}{c}\mu_2=0\end{array}$} }}
  &\multicolumn{2}{|c|}{$\begin{array}{c}\mu_3\ne0\end{array}$ }
    & $\is\,; \ {3\choose 3}\, PEPEP-P$ & 24\rule{0pt}{4.0mm}\\[0.3mm]
\cline{4-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{  }
  &\multicolumn{1}{|c|}{ \raisebox{-0.7em}[0pt][0pt]{$\begin{array}{c}\mu_3=0,\\ \mu_4\ne0\end{array}$} }
  &\multicolumn{1}{|c|}{$  \begin{array}{c}\widetilde M\ne0\end{array}  $  }
   & $\ses; N, \, {4\choose 2}\, PHP-PHP$ & 25\rule{0pt}{4.0mm}\\[0.3mm]
\cline{5-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{  }
  &\multicolumn{1}{|c|}{  }
  &\multicolumn{1}{|c|}{$  \begin{array}{c}\widetilde M=0\end{array}  $  }
   & $ \ses ; {4\choose 3}\,  N $ & 26\rule{0pt}{5.0mm}\\[0.3mm]
\cline{4-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{  }
  &\multicolumn{2}{|c|}{$\begin{array}{c}\mu_3=\mu_4=0,\widetilde M\ne0\end{array}$ }
   & $\big(  \boldsymbol{\ominus}[|];s\big); N,\big(  \boldsymbol{\ominus}[|];N\big)$ & 27\rule{0pt}{4.0mm}\\[0.3mm]
\cline{4-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{  }
  &\multicolumn{1}{|c|}{\multirow{3}*{ \raisebox{-0.0em}[0pt][0pt]{$\begin{array}{c}\mu_3=0,\\ \mu_4=0,\\ \widetilde M=0\end{array}$} }}
   &\multicolumn{1}{|c|}{ $B_6<0$ }
   &  $\big(  \boldsymbol{\ominus}[\parallel^c];\raisebox{0.2em}[0pt][0pt]{${}_\emptyset$}\big);\big(  \boldsymbol{\ominus}[\parallel^c];N\big)$ & 28\rule{0pt}{4.0mm}\\[0.3mm]
\cline{5-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{  }
  &\multicolumn{1}{|c|}{}
   &\multicolumn{1}{|c|}{ $B_6>0$}
   & $\big(  \boldsymbol{\ominus}[\parallel];\raisebox{0.2em}[0pt][0pt]{${}_\emptyset$}\big);\big(  \boldsymbol{\ominus}[\parallel];N\big)$ & 29\rule{0pt}{4.0mm}\\[0.3mm]
\cline{5-7}
   \multicolumn{1}{|@{}c }{}
  &\multicolumn{1}{|c|}{ }
  &\multicolumn{1}{|c|}{  }
  &\multicolumn{1}{|c|}{}
   &\multicolumn{1}{|c|}{ $B_6=0$}
   & $\big(  \boldsymbol{\ominus}[\boldsymbol{|}\text{\scriptsize 2}];\raisebox{0.2em}[0pt][0pt]{${}_\emptyset$}\big);\big(  \boldsymbol{\ominus}[\boldsymbol{|}\text{\scriptsize 2}];N\big)$ & 30\rule{0pt}{4.0mm}\\[0.3mm]
 \hline
\end{tabular}
 \end{center}
\end{table}


\begin{table}\scriptsize
\caption{} \label{table5}
\begin{center}
\begin{tabular}{|@{}c@{ }|@{}c@{}||@{}c@{}|@{}c@{}||@{}c@{}|@{} c@{} |}
 \hhline{|======|}
{  \begin{tabular}{c}\emph{Con-}\\[-0.7mm] \emph{figu-}\\[-0.7mm] \emph{ration} \end{tabular}  } &
  {  \begin{tabular}{c}\emph{Phase}\\[-0.7mm] \emph{portrait} \end{tabular}  } &
{  \begin{tabular}{c}\emph{Con-}\\[-0.7mm] \emph{figu-}\\[-0.7mm] \emph{ration} \end{tabular}  } &
 {  \begin{tabular}{c}\emph{Phase}\\[-0.7mm] \emph{portrait} \end{tabular}  } &
{  \begin{tabular}{c}\emph{Con-}\\[-0.7mm] \emph{figu-}\\[-0.7mm] \emph{ration} \end{tabular}  } &
{  \begin{tabular}{c}\emph{Phase}\\[-0.7mm] \emph{portrait} \end{tabular}  }\rule{0pt}{1.7mm}\\[0.3mm]
 \hhline{|======|}
\raisebox{-2.2em}[0pt][0pt]{1} & \emph{Vul}$_{\,11}$ if $B_1\ne0$ &  8  & \emph{Vul}$_{\,2}$ & 20 &  \emph{Ham}$_{11}$  \rule{0pt}{0.0mm}\\[0.0mm]
 \cline{2- 6}
  & \emph{Vul}$_{\,9}$ if $B_1=0$, $B_3B_4<0$  &  9  & \emph{Vul}$_{\,7}$ &  \raisebox{-0.7em}[0pt][0pt]{21} &\ \emph{Ham}$_{20}$ if $B_3\ne0$  \rule{0pt}{0.0mm}\\[0.0mm]
  \cline{2- 4}\cline{6-6}
  & \emph{Vul}$_{\,8}$ if $B_1=0$, $B_3B_4>0$ &  10  & \emph{Ham}$_{17}$  &   &\ \emph{Ham}$_{21}$ if $B_3=0$ \rule{0pt}{0.0mm}\\[0.0mm]
 \cline{2- 6}
  & \emph{Vul}$_{\,10}$ if $B_1= B_3=0$ &  11  & \emph{Ham}$_{11}$  & 22 & \emph{Vul}$_{\,2}$  \rule{0pt}{0.0mm}\\[0.0mm]
 \hline
\raisebox{-0.7em}[0pt][0pt]{2} & \emph{Ham}$_{26}$ if $B_1=0,B_3B_4<0$  &  12  & \emph{Ham}$_{28}$ & 23 & \emph{Ham}$_{17}$  \rule{0pt}{0.0mm}\\[0.0mm]
 \cline{2- 6}
  & \,\emph{Ham}$_{25}$ if $\Big\{   \begin{array}{l}   B_1\ne0 ,\ or\\[-0.3mm]  B_1=0, B_3B_4>0\end{array}$&  13
     & \emph{Vul}$_{\,2}$ & 24 & \emph{Ham}$_{18}$  \rule{0pt}{0.0mm}\\[0.0mm]
 \hline
\raisebox{-0.7em}[0pt][0pt]{3} & \emph{Ham}$_{27}$ if $B_1\ne0$ &  14   & \emph{Ham}$_{17}$   &  25  & \emph{Ham}$_{13}$   \rule{0pt}{0.0mm}\\[0.0mm]
\cline{2-  6}
  & \emph{Ham}$_{24}$ if $B_1=0$ &  \raisebox{-0.7em}[0pt][0pt]{15}  &\emph{Vul}$_{\,6}$ if $B_1\ne0$  &  26  & \emph{Ham}$_{11}$ \rule{0pt}{0.0mm}\\[0.0mm]
 \cline{1-  2}\cline{4-6}
4 & \emph{Ham}$_{25}$ &    & \emph{Vul}$_{\,5}$ if $B_1=0$  & {27} & \emph{Ham}$_{14}$  \rule{0pt}{4.2mm}\\[0.0mm]
 \cline{1- 6}
5 & \emph{Ham}$_{23}$ &  16  &  \emph{Ham}$_{19}$ & 28  &\,\emph{Ham}$_{11}$   \rule{0pt}{4.2mm}\\[0.0mm]
 \cline{1-  6}
 \raisebox{-0.7em}[0pt][0pt]{6} & \emph{Vul}$_{\,4}$ if $B_1\ne0$&  17 & \emph{Ham}$_{22}$ & 29  &\,\emph{Ham}$_{15}$  \rule{0pt}{4.2mm}\\[0.0mm]
 \cline{2-  6}
  & \emph{Vul}$_{\,3}$ if $B_1=0$& 18   &  \emph{Ham}$_{19}$ & 30   &\,\emph{ Ham}$_{16}$  \rule{0pt}{4.2mm}\\[0.0mm]
 \hline
7 & \emph{Ham}$_{11}$ &  19  &  \emph{Ham}$_{12}$ &  \multicolumn{2}{|@{}c }{}   \rule{0pt}{0.0mm}\\[0.0mm]
  \cline{1-  4}
  \end{tabular}
\end{center}
\end{table}

The proof of Theorem \ref{thm:QS-Hamilt} is based on the
classifications of quadratic Hamiltonian systems given in
\cite{Art-Llib} and \cite{KalVul}. We use the notations of
\cite{Art-Llib} whether the system has a center as $\emph{Vul}_\#$
or if not as $Ham_\#$. In the first paper the global phase
portraits of this family were studied. In the second one there are
determined the affine invariant criteria for the realization of
each one of the 28 possible topologically distinct phase portraits
constructed in \cite{Art-Llib}.  That is, the phase portraits of
the systems
\begin{equation}\label{sys:Ham} {dx\over dt} = {\partial H\over  \partial y},
\quad {dy\over dt} = -{\partial H\over \partial x}, \end{equation} where
$H(x,y)$ is a polynomial of degree 3 in the variables $x$ and $y$
over $\mathbb{R}$.

According to the paper \cite{Art_Llib_Vlp_JDE} for the quadratic
Hamiltonian systems   we have \begin{equation}\label{HamFI:H(x,y)} H(x,y)
=\sum_{j=0}^2 \frac{1}{j+1}{C_j(x,y)}, \end{equation} where $C_j(x,y)$,
$j=0,1,2$ are the polynomials \eqref{expr:Ci,Di}. So the 3rd degree
homogeneous part of the polynomial $H(x,y)$ is the polynomial
$H_3(x,y)=C_2(x,y)/3$. As it is shown in \cite{Art-Llib} via linear
transformations the non-zero form $H_3(x,y)$ can be derived to one of
the 4 canonical forms
$$
  a)\ x(x^2-y^2);\quad b)\ (x^3+y^3)/3;\quad  c)\ x^2y;\quad
d)\ x^3/3.
$$
By \eqref{Expr:Bj} we observe that the invariant polynomials $\eta$
and $\widetilde M$  are respectively the discriminant and the Hessian
of the binary form $C_2(x,y)$.  So considering also \cite{KalVul}  we
arrive to the next proposition.

\begin{proposition}\label{prop:Ham-canon-syst}
Assume that for a quadratic  system
the condition $\frac{\partial}{\partial x}P(x,y)+
\frac{\partial}{\partial y}Q(x,y)\equiv0$   holds, i.e. it is
Hamiltonian.   Then this system could be brought via an affine
transformation and time rescaling  to one of the canonical forms
below if and only if the respective conditions hold
\begin{gather*}
 (S^{(h)}_1)\quad \begin{cases}
 \dot x=\alpha+bx+cy-2xy,\\
 \dot y=\beta-ax-by-3x^2+y^2,\end{cases}
 \Leftrightarrow  \eta>0;\\
(S^{(h)}_2)\quad \begin{cases}
 \dot x= \alpha+bx+cy+y^2,\\
 \dot y=\beta-ax-by-x^2,\end{cases}
 \Leftrightarrow  \eta<0;\\
(S^{(h)}_3)\quad \begin{cases}
  \dot x= \alpha+bx+cy+x^2,\\
  \dot y=\beta-ax-by-2xy,\end{cases}
  \Leftrightarrow   \eta=0,\;  \widetilde M\ne0;\\
(S^{(h)}_4)\quad \begin{cases}
  \dot x= \alpha+bx+cy,\\
  \dot y=\beta-ax-by-x^2,\end{cases}  \Leftrightarrow   \eta=0,
 \; \widetilde M=0.
\end{gather*}
\end{proposition}

\begin{remark}  \label{rem:eta=-27mu0} \rm
For a quadratic Hamiltonian system
the relation $\eta=-27\mu_0$ holds. This could be easy established
via    the frontal evaluations  of the invariant polynomials $\eta$
and $\mu_0$ for systems $(S^{(h)}_1)-(S^{(h)}_4)$.
\end{remark}


\emph{Proof of Theorem \ref{thm:QS-Hamilt}}. In what follows we
shall consider each one of the systems $(S^{(h)}_1)-(S^{(h)}_4)$ and
will show that the conditions  given in \cite{KalVul} for
distinguishing the phase portraits of the respective systems, are
equivalent with the affine invariant provided in Tables \ref{table4}
and \ref{table5}.


\subsection{The family of  systems $(S^{(h)}_1)$}

In this case according to   \cite[Theorem 3]{KalVul}  the phase
portrait (and this yields the respective configuration of
singularities) of a system of the family $(S^{(h)}_1)$ is determined
by the following affine invariant conditions
\begin{equation}\label{cond:ph-port-S1-h}
 \begin{gathered}
  \widehat D < 0,\; \widehat U_c=0,\; \widehat T_c>0
 \Leftrightarrow  \emph{Vul}_8 \Rightarrow   \is,\is,\is,c\,; N,N,N;
\\
  \widehat D < 0,\; \widehat U_c=0,\; \widehat T_c<0
 \Leftrightarrow    \emph{Vul}_9 \Rightarrow   \is,\is,\is,c\,; N,N,N;
\\
  \widehat D < 0,\; \widehat U_c=0,\; \widehat T_c=0
 \Leftrightarrow  \emph{Vul}_{10} \Rightarrow   \is,\is,\is,c\,;N,N,N;
\\
  \widehat D < 0,\; \widehat U_c\ne0  \Leftrightarrow
 \emph{Vul}_{11} \Rightarrow   \is,\is,\is,c\,; N,N,N;
\\
 \widehat  D = 0,\; \widehat U_c=0,\; \widehat T_c=0 \Leftrightarrow
 \emph{Ham}_{23} \Rightarrow  (hhhhhh)_{(4)}\,; N,N,N;
\\
    \widehat  D = 0,\; \widehat U_c=0,\; \widehat T_c<0
 \Leftrightarrow \emph{Ham}_{24} \Rightarrow
 \is,\is,\widehat {cp}_{(2)}\,; N,N,N;
\\
 \begin{cases}\widehat D > 0,\; \widehat U_c\ne0,  \text{ or} \\
    \widehat  D \ge 0,\; \widehat U_c=0,\; \widehat T_c>0
 \end{cases}
 \Leftrightarrow  \emph{Ham}_{25} \Rightarrow
  \begin{cases} \is,\is\,;  N,N,N  &\text{if } \widehat D \ne 0;\\
      \is,\hat s_{(3)}\,;  N,N,N  &\text{if } \widehat D = 0;
\end{cases}
 \\
    \widehat  D > 0,\; \widehat U_c=0,\; \widehat T_c<0
 \Leftrightarrow \emph{Ham}_{26} \Rightarrow  \is,\is\,; N,N,N;
\\
    \widehat  D = 0,\; \widehat U_c\ne0
 \Leftrightarrow  \emph{Ham}_{27} \Rightarrow
 \is,\is,\widehat {cp}_{(2)}\,; N,N,N.
\end{gathered}
\end{equation}

According to \cite{Art-Llib} systems $(S^{(h)}_1)$  have at least one
real singular point. So we may assume   $\alpha=\beta=0 $ due to a
translation and then calculations yield
 \begin{gather}\label{val:S1-Ham}
\widehat  D= 4 (b^2 - a c)^2[-8a^3c -(36c^2 + b^2)a^2 +
6c(4b^2-9c^2)a +3(b^4+33b^2c^2 - 9c^4)], \nonumber \\
\widehat  U_c =  108 bc (a - 2 b - 3 c) (a + 2 b - 3 c)(a - 2 b + c)
 (a + 2 b + c), \nonumber\\
\begin{aligned}
\widehat  {T}_c&= 3456\big[(a+2b+c)^3(a - 2 b - 3 c)^3 +
24bc(a+2b+c)^2(a - 2 b - 3 c)^2\\
 &\quad -384b^2c^2(a+2b+c)(a - 2 b - 3 c)-4096b^3c^3\big],
\end{aligned}\nonumber \\
\mu_0= -12<0,\quad \mathbf{D}=48\widehat  D,\quad B_1=\widehat  U_c/2,\\
B_3= 18 (a^2 - 4 b^2 - 2 a c - 3 c^2) x^3 y -
 18 b c (3 x^4 + 6 x^2 y^2 - y^4),\nonumber \\
 B_4= 288[ (a^2 - 4 b^2 - 2 a c - 3 c^2) x -
   8 b c y][24 b c x + (a^2 - 4 b^2 - 2 a c - 3 c^2) y]. \nonumber
\end{gather}

\begin{remark}  \label{rem:signB3B4} {\rm
We claim that in the case $\widehat
U_c=0$   we have $\operatorname{sign}(\widehat {\,T}_c)=\operatorname{sign}(B_3B_4)$ and $
\widehat {\,T}_c=0$ if and only if $B_3=0$.
} \end{remark}

Indeed assume $\widehat  U_c=0$. We observe that the change of
variables $(x,y,t)\mapsto(x,-y,-t)$ keeps the systems $(S^{(h)}_1)$
with $\alpha=\beta=0 $ except the  sign of the parameter $b$, which
will be changed.  Hence without loss of generality we could consider
only the equality $bc(a+2b+c)(a - 2 b - 3 c)=0$. If $bc\ne0$ then by
$(7.5)$ evidently we get $\widehat {\,T}_c= -2^{19}3^3b^3c^3$. On
the other hand for $a=-2b-c$ as well as for $a=2b+3c$, we obtain
$B_3B_4=-2^{12}3^4b^3c^3(x - y)^4 (3 x + y)^2$ and $B_3=-18 b c (x -
y)^3 (3 x + y)$. So our claim is proved in the considered case.

Assume now $bc=0$. Then calculations yield
$$
\widehat {\,T}_c=3456 (a - 3 c)^3 (a + c)^3,\quad
B_3B_4= 3\widehat {\,T}_c x^4 y^2/2,\quad
B_3=18 (a - 3 c) (a + c) x^3 y
$$
if $b=0$ and
$$
 \widehat{T}_c=3456 (a - 2b)^3 (a + 2b)^3,\quad   B_3B_4=
3\widehat {T}_c x^4 y^2/2,\quad
 B_3=18 (a - 2 b) (a + 2 b) x^3 y
$$
if $c=0$. Now evidently the proof of the claim is completed.

\subsubsection{The case $\widehat  D<0$}
Then $\mathbf{D}<0$ and considering $(7.5)$ and Remark
\ref{rem:signB3B4} we conclude that the conditions for phase
portraits $\emph{Vul}_8-\emph{ Vul}_{11}$ given by Tables
\ref{table4} and \ref{table5} are equivalent to the respective
conditions in \eqref{cond:ph-port-S1-h}.

\subsubsection{The case $\widehat  D>0$}
According to \eqref{cond:ph-port-S1-h} in this case we could have
only the phase portraits $\emph{Ham}_{25}$ and $\emph{Ham}_{26}$. So
considering $(7.5)$ and   Remark \ref{rem:signB3B4} we again arrive
to the equivalence of the respective conditions from
\eqref{cond:ph-port-S1-h} and from Tables \ref{table4} and
\ref{table5}.

\subsubsection{The case $\widehat  D=0$}
By  \eqref{cond:ph-port-S1-h} we have: \emph{(i)} the phase
portraits $\emph{Ham}_{24}$ (if $\widehat U_c=0,\ \widehat T_c<0$)
and $\emph{Ham}_{27}$(if $\widehat U_c\ne0$) with the same
configuration having three finite singularities (one double);
\emph{(ii)} the phase portrait $\emph{Ham}_{25}$ (if $\widehat
U_c=0,\ \widehat T_c>0$) possessing only two real finite
singularities (one triple); \emph{(iii)}  the phase portrait
$\emph{Ham}_{23}$ (if $\widehat U_c= \widehat T_c=0$) possessing
only one finite singularity of multiplicity four. Considering the
diagram in Figure \ref{fig1} and $(7.5)$  we conclude that the
conditions for determining the mentioned phase portraits from Tables
\ref{table4} and \ref{table5} are equivalent to the respective
conditions from \eqref{cond:ph-port-S1-h}.


\subsection{The family of  systems $(S^{(h)}_2)$}

In this case according to   \cite[Theorem 4]{KalVul}  the phase
portrait (and this yields the respective configuration of
singularities) of a system from the family $(S^{(h)}_2)$ is
determined by the following affine invariant conditions
\begin{equation}\label{cond:ph-port-S2-h}
\begin{gathered}
  \begin{cases} \widehat D > 0,  \text{ or} \\
  \widehat  D=\widehat  T=\widehat  P=0, \;\widehat  R\ne 0
\end{cases}
 \Leftrightarrow  \emph{Vul}_2 \Rightarrow
\begin{cases}
\is,c\,;  N  & \text{if } \widehat D \ne 0;\\
 c,\hat s_{(3)}\,; N  & \text{if } \widehat D = 0;
\end{cases}
\\
 \widehat D < 0,\;  \widehat R>0,\widehat S>0,  \widehat U_c=0
  \Leftrightarrow \emph{Vul}_3 \Rightarrow  \is,\is,c,c\,; N;
\\
 \widehat D < 0, \; \widehat R>0,\widehat S>0,  \widehat U_c\ne0
 \Leftrightarrow  \emph{Vul}_4 \Rightarrow   \is,\is,c,c\,; N;
\\
 \widehat D = 0,\;  \widehat T<0
 \Leftrightarrow  \emph{Vul}_7 \Rightarrow
  \is,c,\widehat {cp}_{(2)}\,; N;
\\
 \begin{cases}
 \widehat D < 0, (\widehat R\le0)\vee(\widehat S \le0),  \text{ or} \\
 \widehat  D=\widehat  T=0,\; \widehat  P \widehat  R< 0
\end{cases}
 \Leftrightarrow  \emph{Ham}_{11} \Rightarrow  \ses;  N;
\\
\begin{cases}
\widehat D = 0,\;  \widehat T>0,  \text{ or} \\
\widehat  D=\widehat  T=\widehat  P=  \widehat  R=0
\end{cases}
 \Leftrightarrow  \emph{Ham}_{17} \Rightarrow
\begin{cases}
\widehat {cp}_{(2)}\,; N  & \text{if }\widehat T \ne 0;\\
(hh)_{(4)}\,;\, N  & \text{if } \widehat T = 0;
\end{cases}
  \\
 \widehat  D=\widehat  T=0,\; \widehat  P \widehat  R> 0
  \Leftrightarrow  \emph{Ham}_{28} \Rightarrow
  \widehat {cp}_{(2)},\widehat {cp}_{(2)}\,;  N.
\end{gathered}
\end{equation}
Taking into account  \eqref{comit:Ham-Sys} and
\eqref{Pol:D,T,R,e.al.} a straightforward calculation gives for
systems $(S^{(h)}_2)$ \begin{equation}\label{val:S2-Ham}
\begin{aligned} \mu_0=1,\ \widehat D&=  \mathbf{D}/48, \ \widehat
T=\mathbf{T}/6,\   \widehat R=\mathbf{R}/4,\   \widehat S&=
\mathbf{S}/48, \ \widehat  P&=  \mathbf{P}, \ \widehat U_c&= 2B_1.
\end{aligned} \end{equation} Herein considering the diagram from
Figure 1, it is easy to observe that the conditions for determining
the phase portraits from Tables \ref{table4} and \ref{table5}
corresponding to the case $\mu_0>0$ are equivalent to the respective
conditions from \eqref{cond:ph-port-S2-h}.


\subsection{The family of  systems $(S^{(h)}_3)$}

For this family of systems we have $\eta=\mu_0=0$ and $\widetilde
M\ne0$. According to \cite[Table \ref{table2}]{Dana-Vlp-JDE}
at infinity we
have two real singularities of total multiplicity at least four. So
in order to determine exactly the configuration  of the singularities
at infinity we shall use the  classification of the behavior of the
trajectories in the neighbourhood  at infinity of quadratic
differential systems, given in \cite{Dana-Vlp-JDE}.

In this order of ideas we need the following additional invariant
polynomials, defined in \cite{Dana-Vlp-JDE}
\begin{equation}\label{comit:infty}
\begin{gathered}
\kappa(\tilde a)    =   (\widetilde M,\widetilde K)^{(2)},\ \
    \kappa_1(\tilde a)   =  (\widetilde M,C_1)^{(2)},\\
   \widetilde K_1(\tilde a,x,y)    =   p_1(x,y)q_2(x,y)-p_2(x,y)q_1(x,y),\\
    \widetilde  K_2(\tilde a,x,y)   =   4(T_2,\omega)^{(1)} +3
     D_1(C_1,\omega)^{(1)}  -\omega\big(16T_3+3T_4/2+3D_1^2\big),\\
 \widetilde K_3(\tilde a,x,y)= C_2^2(4T_3+3T_4) +C_2(3C_0\widetilde
 K - 2C_1T_7)+\widetilde K_1(3\widetilde K_1-C_1D_2),
\end{gathered}
\end{equation}
where $\omega=\widetilde M-8\widetilde K$.  According to
\cite[Theorem 2]{KalVul} the phase portrait of a system from the
family $(S^{(h)}_3)$ is determined by the following affine invariant
conditions
\begin{equation}\label{cond:ph-port-S3-h}
\begin{gathered}
  \emph{Vul}_5   \Leftrightarrow  \widehat D < 0,\;  \widehat U_c=0;\\
  \emph{Vul}_6   \Leftrightarrow  \widehat D < 0,\;  \widehat U_c\ne0;\\
 \emph{Ham}_{12}   \Leftrightarrow  \widehat R = 0,\;  \widehat U<0 ;\\
 \emph{Ham}_{13}   \Leftrightarrow  \widehat R =   \widehat U=0,\  \widehat V\ne0;\\
 \emph{Ham}_{19}   \Leftrightarrow
 \begin{cases}
\widehat D > 0,   \text{ or} \\
\widehat  D=0,\;  \widehat  R\ne 0,\;\widehat U_c=0;
\end{cases}
\\
 \emph{Ham}_{20}   \Leftrightarrow  \widehat R = 0,\;
      \widehat U>0,\  \widehat W_2\ne0 ;\\
 \emph{Ham}_{21}   \Leftrightarrow  \widehat R = 0,\;
      \widehat U>0,\;  \widehat W_2=0 ;\\
 \emph{Ham}_{22}   \Leftrightarrow  \widehat  D=0,\; \widehat  U_c\ne0.
\end{gathered}
\end{equation}
For systems $(S^{(h)}_3)$ calculations yield
\begin{equation}\label{val:S3-Ham}
\begin{gathered}
 \widehat  D =108 c^4 (a b + 2 \beta)^2 - 4 c^2 (b^2
 + 2a c - 4 \alpha)^3, \\
\widehat U_c = -108 c^4 (a b + 2 \beta),\quad
 \widehat  R =12 c^2 x^2,\\
 \mu_0=0,\quad  \widetilde M= -72 x^2,\quad
 \mathbf{D}=48\widehat  D,\quad B_1=\widehat U_c/2, \\
\mu_1=4cx,\quad \tilde L=24 x^2>0,\quad \widetilde K=-4x^2<0,
\end{gathered}
\end{equation}

\subsubsection{The case $\widehat  R\ne0$} In this   case the
condition $c\ne0$ (i.e., $\mu_1\ne0$) holds and as $\tilde L>0 $ and
$ \widetilde K<0$ according to \cite[Table 4]{Dana-Vlp-JDE} at
infinity the behavior of the trajectories corresponds to Figure 9;
i.e., we have the following configuration of singularities $N,
\widehat {1\choose 2}PEP-H$.

We observe that in the case $\widehat D=0$ and
(i.e., $\mathbf{D}=0$) by
\eqref{cond:ph-port-S3-h} we have the phase portrait
$\emph{Ham}_{19}$ with one finite singularity (if $\widehat
U_c=0$) and $\emph{Ham}_{22}$ with two finite singularities  (if
$\widehat U_c\ne0$). Considering the diagram (Figure \ref{fig1}),
\eqref{val:S3-Ham}  and the fact, that the condition $\widehat
D^2+ \widehat U_c^2\ne0$ implies $\widehat R\ne0$, we conclude
that the conditions for determining the phase portraits from
Tables \ref{table4} and \ref{table5} corresponding to the
case $\mu_0=0$ and $\mu_1\ne0$
are equivalent to the respective conditions of
\eqref{cond:ph-port-S3-h} (i.e. the conditions for the phase
portraits $\emph{Vul}_5, \emph{Vul}_6$, $\emph{Ham}_{19}$ and
$\emph{Ham}_{22}$).

\subsubsection\emph{The case $\widehat  R=0$}

Then $c=0$ and  for systems $(S^{(h)}_3)$ we calculate
\begin{equation}\label{val:S3-Ham-c0}
\begin{gathered}
\widehat U= (b^2 - 4 \alpha) \big[(a
b+2\beta) x - (b^2   - 4 \alpha)y \big]^2x^4, \quad
\widehat W_2=-6 x^4 (a b + 2 \beta),\\
\mu_0=\mu_1=0,\ \mu_2=  (4\alpha-b^2)x^2,\quad
 \mathbf{U}=\widehat U,\quad B_3=3\widehat W_2,\\
\kappa=\kappa_1=0,\quad \widetilde K_2=768(b^2 - 4 \alpha) x^2.
\end{gathered}
\end{equation}
We observe that $\operatorname{sign}(\widehat U)
=\operatorname{sign}(\mathbf{U})=-\operatorname{sign}(\mu_2)
=\operatorname{sign}(\widetilde K_2)$.

\paragraph{Assume first $\widehat U<0$}
According to \eqref{cond:ph-port-S3-h}
the phase portrait of a system $(S^{(h)}_3)$ corresponds to
$\emph{Ham}_{12}$ (without real finite singularities). On the other
hand as $\mu_0=\mu_1=\kappa=\kappa_1=0$, $\mu_2>0$, $\tilde L>0
$ and $ \widetilde K_2<0$, according to \cite[Table \ref{table4}]{Dana-Vlp-JDE}
at infinity the behavior of the trajectories corresponds to Figure 8,
i.e. we arrive to the configuration  $ N, {2\choose 2} H-H$.

\paragraph{Admit now $\widehat U>0$}  Then by
\eqref{cond:ph-port-S3-h} we obtain the phase portrait
$\emph{Ham}_{20}$ if $\widehat W_2\ne0$ and $\emph{Ham}_{21}$ if
$\widehat W_2=0$ in  both cases  having two finite integrable
saddles. As regarding the configuration of infinite singularities we
observe that $\mu_2<0$, $\tilde L>0 $ and $ \widetilde K <0$. So
following \cite[Table \ref{table4}]{Dana-Vlp-JDE} we obtain Figure 9, i.e. we
get the configuration   $  N, {2\choose 2}
P\;E\;P-P\;E\;P$. This leads to the total
configuration  21 of  Table \ref{table4}.  It remains to note that by
\eqref{val:S3-Ham-c0} the condition $\widehat W_2=0$ is equivalent to
$B_3=0$.

\paragraph{Assume finally $\widehat U=0$} Then
$\alpha=b^2/4$ and for systems $(S^{(h)}_3)$ we have
\begin{equation}\label{val:S3-Ham-c0-U0}
 \mu_0=\mu_1=\mu_2=\mu_3=0,\ \mu_4= (a b + 2 \beta)^2x^4/4
= \widehat V.
\end{equation}
According to \eqref{cond:ph-port-S3-h} the phase portrait
of a system $(S^{(h)}_3)$ corresponds to $\emph{Ham}_{13}$ if
$\widehat V\ne0$ and we get   degenerate systems (with the phase
portrait  $\emph{Ham}_{14}$) if $\widehat V=0$. We claim that in
the first case at infinity we have the configuration $N, \,
{4\choose 2}\, PHP-PHP$. Indeed following \cite{Dana-Vlp-JDE} for
systems $(S^{(h)}_3)$  we have
$$
 \kappa=\kappa_1=0,\quad \tilde L=24 x^2>0,\quad
 \widetilde K=-4x^2<0, \quad   \widetilde R=-8x^2<0
$$
and according to  \cite[Table \ref{table4}]{Dana-Vlp-JDE} at
infinity the behavior of the trajectories corresponds to
Figure 28; i.e., we arrive
to the mentioned above configuration and our claim is proved.

It remains to observe that in the case $\widehat V=0$ (then
$\mu_4=0$) and the degenerate systems have the phase portrait
$\emph{Ham}_{14}$. Hence the singular invariant line coincides with
the invariant line of the respective linear systems,  and using our
notations (see page \pageref{pageref:notat}) we get the
configuration $\big(  \boldsymbol{\ominus}[|];s\big); N,\big(
\boldsymbol{\ominus}[|];N\big)$.

Considering that the condition $\widetilde M\ne0$ for the family
$(S^{(h)}_3)$ holds  and \eqref{val:S3-Ham-c0}, we conclude that the
conditions for determining the phase portraits from Tables \ref{table4} and \ref{table5}
corresponding to the case $\mu_0=\mu_1=0$   are equivalent to the
respective conditions from \eqref{cond:ph-port-S3-h} (i.e. the
conditions for the phase portraits $\emph{Ham}_{12},
\emph{Ham}_{13}$, $\emph{Ham}_{20}$ and $\emph{Ham}_{21}$).

\subsection{The family of  systems $(S^{(h)}_4)$}
For this family of systems we have $\eta=\mu_0=\mu_1=0$ and
$\widetilde M=0$. According to \cite[Table \ref{table2}]{Dana-Vlp-JDE}
at infinity we have one
real singularity of total multiplicity at least five. So in order to
determine exactly the configuration  of the singularities at
infinity we shall use again Table \ref{table4} from \cite{Dana-Vlp-JDE}.

According to \cite[Theorem 1]{KalVul} the phase portrait  of a system
from the family $(S^{(h)}_4)$ is determined by the following affine
invariant conditions
\begin{equation}\label{cond:ph-port-S4-h}
\begin{gathered}
   \emph{Vul}_2   \Leftrightarrow  \widehat P\ne0,\  \widehat U>0;\\
 \emph{Ham}_{11}   \Leftrightarrow
 \begin{cases}
\widehat P\ne0,\; \widehat U<0,  \text{ or} \\
\widehat  P= \widehat  U=0,\;  \widehat V\ne0, \text{ or}\\
\widehat  P= \widehat  U= \widehat V=0, \widehat W_1<0;
\end{cases}
\\
 \emph{Ham}_{15}   \Leftrightarrow  \widehat  P= \widehat
 U= \widehat V=0, \widehat W_1>0;\\
 \emph{Ham}_{16}   \Leftrightarrow  \widehat  P= \widehat
 U= \widehat V=\widehat W_1=0;\\
 \emph{Ham}_{17}   \Leftrightarrow  \widehat  P\ne0,\; \widehat U=0;\\
 \emph{Ham}_{18}   \Leftrightarrow  \widehat  P=0,\ \widehat U\ne0.\\
\end{gathered}
\end{equation}
For systems $(S^{(h)}_4)$ calculations yield
\begin{equation}\label{val:S4-Ham}
\begin{gathered}
 \widehat  P =c^4 x^4,\quad
\widehat U  = \big[ (ac-b^2)^2+4c(b\alpha+c\beta)\big](bx+cy)^2x^4,\\
  \eta=\widetilde M=  \mu_0=\mu_1=0, \quad
 \mu_2=c^2 x^2, \quad  \widetilde K= 0,\quad \mathbf{P}=\widehat P,\quad
  \mathbf{U}=\widehat U.
\end{gathered}
\end{equation}

\subsubsection{The case $\widehat  P\ne0$}
In this   case according to  \eqref{cond:ph-port-S4-h}    a system
from the family $(S^{(h)}_4)$ has  one of the following phase
portraits: $\emph{Vul}_2$ (if $\widehat U>0$), $\emph{Ham}_{11}$ (if
$\widehat U<0$) or $\emph{Ham}_{17}$ (if $\widehat U=0$). In all the
cases at infinity we have a multiple node, which according to
\cite[Table \ref{table4}]{Dana-Vlp-JDE} is of multiplicity five, as
the conditions $\mu_0=\mu_1= \widetilde M=\widetilde K=0$ and
$\mu_2>0$ (as $c\ne0$)  are verified. More precisely at infinity we
have Figure 30, i.e. the nilpotent singular point $
\widehat{2\choose 3}\, N $.

\subsubsection{The case $\widehat  P=0$} Then $c=0$ and for
systems $(S^{(h)}_4)$ we have
\begin{equation}\label{val:S4-Ham-c0}
\begin{gathered}
\widehat U  =  b^6x^6,\quad
 \widehat  V=  \big[(\alpha^2-ab\alpha-b^2\beta)x
- b^2\alpha y\big]x^3,\\
  \mu_2=0, \quad \mu_3=b^3 x^3,\quad
  \mathbf{U}=\widehat U,\quad \mu_4=\widehat V,\quad
  \widetilde K= 0,\quad \widetilde   K_1=-bx^3.
\end{gathered}
\end{equation}
So if $\widehat U\ne0$ (i.e., $b\ne0$) according to
\eqref{cond:ph-port-S4-h} we get the phase portrait
$\emph{Ham}_{18}$ with a finite integrable saddle. On the other
hand as $\mu_3\widetilde  K_1=-b^4x^4<0$ according to
\cite[Table 4]{Dana-Vlp-JDE} the configuration of infinite
singularities corresponds to Figure 33; i.e.,
 ${3\choose 3}\, PEPEP-P$.

Assume $\widehat U=0$ (i.e., $b=0$). In this case we have
\begin{equation}\label{val:S4-Ham-c0-A}
 \mu_3=0, \quad \mu_4=\widehat V=\alpha^2x^4,\quad
 \widehat W_1=(a^2+4\beta)x^4-4\alpha x^3y=B_6,\quad
 \widetilde K_3=0
\end{equation}
and we shall consider two subcases: $\mu_4\ne0$ and $\mu_4=0$.
\smallskip

\textbf{(1)} If $\mu_4\ne0$ systems $(S^{(h)}_4)$ are non-degenerate
having the phase portrait $\emph{Ham}_{11}$ (see
\eqref{cond:ph-port-S4-h}, as $\widehat V\ne0$). Clearly at infinity
we have a singular point of multiplicity seven. As $\mu_4>0$ and
$\widetilde K_3=0$ according to \cite[Table \ref{table4}]{Dana-Vlp-JDE} at
infinity we get Figure 30. More exactly we have the configuration
${4\choose 3}\,P-P$ as the infinite singular point is intricate.
\smallskip

\textbf{(2)} Assume finally  $\mu_4=0$, i.e. $\alpha =0$. Then
systems $(S^{(h)}_4)$ with $c=b=\alpha=0$ become degenerate
possessing the phase portraits indicated respectively in
\eqref{cond:ph-port-S4-h}.  We observe that in the case of  all the
three phase portraits $\emph{Ham}_{11}$, $\emph{Ham}_{15}$ and
$\emph{Ham}_{16}$ the systems $(S^{(h)}_4)$ have an invariant
singular conic which is reducible. More exactly, it splits into two
parallel invariant lines which are real and distinct if $B_6>0$
(portrait $\emph{Ham}_{15}$); they are complex if $B_6<0$ (portrait
$\emph{Ham}_{11}$), and they coincide if $B_6=0$ (portrait
$\emph{Ham}_{16}$). This leads to the respective configurations of
singularities described in Table \ref{table4} (see rows No. 28--30).

 It remains to
note taking into account \eqref{val:S4-Ham}, \eqref{val:S4-Ham-c0}
and \eqref{val:S4-Ham-c0-A}  that for the family of systems
$(S^{(h)}_4)$   the respective conditions for determining the
phase portraits from Tables \ref{table4} and \ref{table5} in all the cases considered
above  are equivalent to the respective conditions from
\eqref{cond:ph-port-S4-h}.

As all the cases are considered Theorem \ref{thm:QS-Hamilt} is
proved.


\subsection*{Acknowledgements}
The first two authors are partially supported by grants MTM
2008-03437 from  MICINN/FEDER, and 2009SGR 410 from AGAUR. The
second author also is supported by ICREA Academia.


\begin{thebibliography}{99}

\bibitem{Art-Llib} {J.~C. Art\'es and 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 and 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{Art-Llib-Vlp-IJBCh} {J.~C. Art\'es, J. Llibre and N.~I. Vulpe};
\emph{Singular points of quadratic systems: A complete
classification in the coefficient space $\mathbb{R}^{12}$},
International J. of Bifurcation and Chaos \textbf{18} (2008),
{313--362}.

\bibitem{Art_Llib_Vlp_JDE} {J.~C. Art\'es, J. Llibre and N.~I. Vulpe};
\emph{Quadratic systems with a polynomial first integral:  a
complete classification in the coefficient space $\mathbb{R}^{12}$,} J.
Differential Equations \textbf{246} (2009), 3535--3558.


\bibitem{Blt-Vlp-BASM} {V. A. Baltag and  N. I.  Vulpe};
\emph{Affine-invariant conditions for determining the number and
multiplicity of singular points of quadratic differential systems},
Izv. Akad. Nauk Respub. Moldova Mat. (1993), no. 1, 39--48

\bibitem{Blt-Vlp-DEDS} {V. A. Baltag and N. I. Vulpe},
\emph{Total multiplicity of all finite critical points of the
polynomial differential system}, Differential Equations \& Dynam.
Systems \textbf{5} (1997), 455--471.

\bibitem{Bautin:1962} {N.~N. Bautin};
\emph{On the number of limit cycles which appear with the variation of
the coefficients from an equilibrium position of focus or center
type}, Translations of the Amer. Math. Soc. \textbf{1} (1954), 396--413.

\bibitem{Bul-Tim} {D. Bularas, Iu. Calin. L. Timochouk and  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;\\
(URL: ftp://ftp.its.tudelft.nl/publications/tech-reports/1996
/DUT-TWI-96-90.ps.gz).

\bibitem{Chava} {J. Chavarriga, H. Giacomini, J. Gin\'e and J. Llibre};
\emph{ On the integrability of two--dimensional flows,} J.
Differential Equations \textbf{157} (1999), 163--182.

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

\bibitem{Dum-Llib-Art} {F. Dumortier, J. Llibre and J.~C. Art\'es};
\emph{Qualitative Theory of Planar Differential Systems},
Universitext, Springer--Verlag, New York--Berlin, 2006.


\bibitem{J_L}  {Jiang, Q. and Llibre, J.};
 \emph{Qualitative Theory of singular points},
\textbf{6}, (2005), 87--167.

\bibitem{Kal90} {Yu. F. Kalin};
\emph{Conditions for qualitative pictures with centers for a
complete quadratic system with {$I_9=0,$} {$K_1\not\equiv 0$}}, Izv.
Akad. Nauk Moldav. SSR Mat. (1990), No. 3, 68--71.

\bibitem{Kal92} {Yu. F. Kalin};
\emph{Conditions for the topological distinguishability of complete
quadratic systems with a center for {$K_1\equiv 0$}}, Izv. Akad.
Nauk Moldav. SSR Mat.  (1992), No. 3, 43--45.

\bibitem{KalSib} {Yu. F. Kalin and K. S. Sibirski{\u\i}};
\emph{Conditions for the existence of qualitative portraits with
centers of a complete quadratic system with {$I_9\neq 0$}}, Izv.
Akad. Nauk Moldav. SSR Mat.  (1990), No. 1, 17--26.

\bibitem{KalVul} {Yu. F. Kalin and N. I.  Vulpe};
\emph{Affine-invariant conditions for the topological
discrimination of quadratic {H}amiltonian differential systems},
Differ. Uravn. \textbf{34} (1998), No. 3, 298--302.

\bibitem{Li} {M. A. Liapunov};
\emph{Probl\`eme g\'en\'eral de la stabilit\'e du mouvement}, Ann. of
Math. Stud. \textbf{17}, Princeton University Press, 1947.

\bibitem{Ll} {J. Llibre};
\emph{On the integrability of the differential systems in dimension
two and of the polynomial differential systems in arbitrary
dimension},   Journal of Applied  Analysis and Computation \textbf{1}
(2011), {33--52}.


\bibitem{Llibre-Schlomiuk:2004} {J. Llibre and 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{Mos} {R. Moussu};
\emph{Une d\'emonstration d'un th\'eor\`eme de Lyapunov--Poincar\'e},
Ast\'erisque \textbf{98-99} (1982), 216--223.

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

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


\bibitem{Poincare:1891} {H. Poincar\'e};
\emph{Sur l'int\'egration des \'equations
diff\'erentielles du premier ordre et du premier degr\'e I and II},
Rendiconti del circolo matematico di Palermo \textbf{5} (1891),
161--191; \textbf{11} (1897), 193--239.

\bibitem{Reyn} {J. W. Reyn};
 \emph{A bibliography of the qualitative theory of
quadratic systems of differentiel equations in the plane}, Third
edition, Report 94-02, Delft University of Technology, Faculty of
Technical Mathematics and Informatics, 1994.

\bibitem{Ro} {R. Roussarie};
\emph{Bifurcation of planar vector fields and Hilbert's sixteenth
problem}, Progress in Mathematics, Vol. \textbf{164}, Birkh\"{a}user
Verlag, Basel, 1998.

\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{Pal-Schlomiuk:2001a} {D. Schlomiuk and J. Pal};
\emph{On the geometry in the neighborhood of infinity of quadratic
differential phase portraits with a weak focus}, {Qualitative
Theory of Dynamical Systems}  \textbf{2} (2001), {1--43}.

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

\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{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{Vlp_prepr_B-2010}  {N. Vulpe};
\emph{Characterization of the finite weak singularities of
quadratic systems via invariant theory}.  Preprint 23, Barcelona,
2010, 42 pp.

\bibitem{Ye} {Ye Yanqian};
\emph{Theory of limit cycles}, Trans. of Mathematical Monographs
\textbf{66}, Amer. Math. Soc., Providence, RI, 2 edition, 1984.

\bibitem {Ye2} {Ye Yanqian};
\emph{Qualitative Theory of Polynomial Differential Systems},
Shanghai Scientific \& Technical Publishers, Shanghai, 1995 (in
Chinese).

\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}

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