Global Configurations of Singularities for Quadratic Differential Systems with Total Finite Multiplicity Three and at Most Two Real Singularities

In this work we consider the problem of classifying all configurations of singularities, finite and infinite, of quadratic differential systems, with respect to the geometric equivalence relation defined in Artés et al. (Rocky Mount J Math, 2014). This relation is deeper than the topological equivalence relation which does not distinguish between a focus and a node or between a strong and a weak focus or between foci of different orders. Such distinctions are however important in the production of limit cycles close to the foci in perturbations of the systems. The notion of geometric equivalence relation of configurations of singularities allows us to incorporate all these important geometric features which can be expressed in purely algebraic terms. This equivalence relation is also deeper than the qualitative equivalence relation introduced in [15]. The geometric classification of all configurations of singularities, finite and infinite, of quadratic systems was initiated in [3] where the classification was done for systems with total multiplicity mf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_f$$\end{document} of finite singularities less than or equal to one. That work was continued in [4] where the geometric classification was done for the case mf=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_f=2$$\end{document}. The case mf=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_f=3$$\end{document} has been split in two separate papers because of its length. The subclass of three real distinct singular points was done in [5] and we complete this case here. In this article we obtain geometric classification of singularities, finite and infinite, for the remaining three subclasses of quadratic differential systems with mf=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_f=3$$\end{document} namely: (i) systems with a triple singularity (19 configurations); (ii) systems with one double and one simple real singularities (62 configurations) and (iii) systems with one real and two complex singularities (75 configurations). We also give here the global bifurcation diagrams of configurations of singularities, both finite and infinite, with respect to the geometric equivalence relation, for these subclasses of systems. The bifurcation set of this diagram is algebraic. The bifurcation diagram is done in the 12-dimensional space of parameters and it is expressed in terms of invariant polynomials. This provides an algorithm for computing the geometric configuration of singularities for any quadratic system in this class.


Introduction and Statement of Main Results
We consider here differential systems of the form where p, q ∈ R[x, y], i.e. p, q are polynomials in x, y over R. We call degree of a system (1) the integer m = max(deg p, deg q). In particular we call quadratic a differential system (1) with m = 2. We denote here by QS the whole class of real quadratic differential systems. The study of the class QS has proved to be quite a challenge since hard problems formulated more than a century ago, are still open for this class. It is expected that we have a finite number of phase portraits in QS. We have phase portraits for several subclasses of QS but to obtain the complete topological classification of these systems, which occur rather often in applications, is a daunting task. This is partly due to the elusive nature of limit cycles and partly to the rather large number of parameters involved. This family of systems depends on twelve parameters but due to the group action of real affine transformations and time homotheties, the class ultimately depends on five parameters which is still a rather large number of parameters. For the moment only subclasses depending on at most three parameters were studied globally, including global bifurcation diagrams (for example [1]). On the other hand we can restrict the study of the whole quadratic class by focusing on specific global features of the systems in this family. We may thus focus on the global study of singularities and their bifurcation diagram. The singularities are of two kinds: finite and infinite. The infinite singularities are obtained by compactifying the differential systems on the sphere, on the Poincaré disk or on the projective plane as defined in Sect. 2 (see [13,16]).
The global study of quadratic vector fields began with the study of these systems in the neighborhood of infinity [11,18,22,23]. In [6] the authors classified topological (adding also the distinction between nodes and foci) the whole quadratic class, according to configurations of their finite singularities.
To reduce the number of phase portraits in half in topological classifications problems of quadratic systems, the topological equivalence relation was taken to mean the existence of a homeomorphism of the phase plane carrying orbits to orbits and preserving or reversing the orientation.
We use the concepts and notations introduced in [2,3] which we describe in Sect. 2. To distinguish among the foci (or saddles) we use the notion of order of the focus (or of the saddle) defined using the algebraic concept of Poincaré-Lyapunov constants. We call strong focus (or strong saddle) a focus (or a saddle) whose linearization matrix has non-zero trace. Such a focus (or saddle) will be denoted by f (respectively s). A focus (or saddle) with trace zero is called a weak focus (weak saddle).
We denote by f (i) (s (i) ) the weak foci (weak saddles) of order i, by $ the integrable saddles and by c (respectively c ; c ⊕ ) the centers (respectively isochronous centers; uniform isochronous centers). For more notations see Sect. 2.5.

Definition 1 A center O of a system (1) is isochronous if the period function of solution curves in a punctured neighbourhood V \{O} of O is constant.
Definition 2 Let us consider a system (1) possessing a center, which we may assume to be placed at the origin, having its corresponding linear part as −y, x. We say that this center is uniformly isochronous if in polar coordinates x = ρ cos(ϕ), y = ρ sin(ϕ), this system takes the formρ = α(ρ, ϕ),φ = C, where C is a constant which we may assume to be 1.
In the topological classification no distinction was made among the various types of foci or saddles, strong or weak of various orders. However these distinctions of an algebraic nature are very important in the study of perturbations of systems possessing such singularities. Indeed, the maximum number of limit cycles which can be produced close to the weak foci in perturbations depends on the orders of the foci.
There are also three kinds of simple nodes: nodes with two characteristic directions (the generic nodes), nodes with one characteristic direction and nodes with an infinite number of characteristic directions (the star nodes). The three kinds of nodes are distinguished algebraically. Indeed, the linearization matrices the nodes of two directions have distinct eigenvalues, they have identical eigenvalues and they are not diagonal for the one direction nodes and they have identical eigenvalues and they are diagonal for the star nodes (see [2][3][4]). We recall that the star nodes and the one direction nodes could produce foci in perturbations.
Furthermore, a generic node at infinity may or may not have the two exceptional curves lying on the line at infinity. This leads to two different situations for the phase portraits. For this reason we split the generic nodes at infinity in two types as indicated in Sect. 2.5.
The geometric equivalence relation (see further below) for finite or infinite singularities, introduced in [2] and used in [3][4][5], takes into account such distinctions. This equivalence relation is also deeper than the qualitative equivalence relation introduced by Jiang and Llibre in [15] because it distinguishes among the foci (or saddles) of different orders and among the various types of nodes. This equivalence relation induces also a deeper distinction among the more complicated degenerate singularities.
In quadratic systems these could be of orders 1, 2 or 3 [8]. For details on Poincaré-Lyapunov constants and weak foci of various orders we refer to [16,21]. As indicated before, algebraic information plays a fundamental role in the study of perturbations of systems possessing such singularities. In [26] necessary and sufficient conditions for a quadratic system to have weak foci (saddles) of orders i, i = 1, 2, 3 are given in invariant form.
For the purpose of classifying QS according to their singularities, finite or infinite, we use the geometric equivalence relation which involves only algebraic methods. It is conjectured that there are over 1,000 distinct geometric configurations of singularities. The first step in this direction was done in [2] where the global classification of singularities at infinity of the whole class QS, was done according to the geometric equivalence relation of configurations of infinite singularities. This work was then partially extended to also incorporate finite singularities. We initiated this work in [3] where this classification was done for the case of singularities with a total finite multiplicity m f ≤ 1, continued it in [4] where the classification was done for m f = 2 and in [5] where the classification was done for 3 distinct real finite singularities.
In the present article our goal is to go one step further in the geometric classification of global configurations of singularities by studying here the case of finite singularities with total finite multiplicity three and at most two real finite singularities.
We recall below the notion of geometric configuration of singularities defined in [4] for both finite and infinite singularities. We distinguish two cases: 1) If we have a finite number of infinite singular points and a finite number of finite singularities we call geometric configuration of singularities, finite and infinite, the set of all these singularities each endowed with its own multiplicity together with the local phase portraits around real singularities endowed with additional geometric structure involving the concepts of tangent, order and blow-up equivalences defined in Section 4 of [2] and using the notations described here in Sect. 2.5. 2) If the line at infinity Z = 0 is filled up with singularities, in each one of the charts at infinity X = 0 and Y = 0, the corresponding system in the Poincaré compactification (see Sect. 2) is degenerate and we need to do a rescaling of an appropriate degree of the system, so that the degeneracy be removed. The resulting systems have only a finite number of singularities on the line Z = 0. In this case we call geometric configuration of singularities, finite and infinite, the set of all points at infinity (they are all singularities) in which we single out the singularities at infinity of the "reduced" system, taken together with their local phase portraits and we also take the local phase portraits of finite singularities each endowed with additional geometric structure to be described in Sect. 2.

Remark 1
We note that the geometric equivalence relation for configurations is much deeper than the topological equivalence. Indeed, for example the topological equivalence does not distinguish between the following three configurations which are geometrically non-equivalent: 1) n, f ; 1 1 S N, ©, ©, 2) n, f (1) ; 1 1 S N, ©, ©, and 3) n d , f (1) ; S N, ©, © where n and n d mean singularities which are nodes, respectively two directions and one direction nodes, capital letters indicate points at infinity, © in case of a complex point and S N a saddle-node at infinity and 1 1 encodes the multiplicities of the saddle-node S N . For more details on the notation see Sect. 2.5.
The invariants and comitants of differential equations used for proving our main result are obtained following the theory of algebraic invariants of polynomial differential systems, developed by Sibirsky and his disciples (see for instance [7,9,20,25,27] (C) The Figs. 1, 2, 3 actually contain the global bifurcation diagrams in the 12dimensional space of parameters, of the global geometric configurations of singularities, finite and infinite, of these subclasses of quadratic differential systems and provide an algorithm for finding for any given system in any of the three families considered, its respective geometric configuration of singularities.

Compactification on the Sphere and on the Poincaré Disk
Planar polynomial differential systems (1) can be compactified on the 2-dimensional sphere as follows. We first include the affine plane (x, y) in R 3 , with its origin at (0, 0, 1), and we consider it as the plane Z = 1. We then use a central projection to send the vector field to the upper and to the lower hemisphere. The vector fields thus  obtained on the two hemispheres are analytic and diffeomorphic to our vector field on the (x, y) plane. By a theorem stated by Poincaré and proved in [14] there exists an analytic vector field on the whole sphere which simultaneously extends the vector fields on the two hemispheres to the whole sphere. We call Poincaré compactification on the sphere of the planar polynomial system, the restriction of the vector field thus obtained on the sphere, to the upper hemisphere completed with the equator. For more details we refer to [13]. The vertical projection of this vector field defined on the upper hemisphere and completed with the equator, yields a diffeomorphic vector field on the unit disk, called the Poincaré compactification on the disk of the polynomial differential system. By a singular point at infinity of a planar polynomial vector field we mean a singular point of the vector field which is located on the equator of the sphere, also located on the boundary circle of the Poincaré disk.

Compactification on the Projective Plane
For a polynomial differential system (1) of degree m with real coefficients we associate the differential equation ω 1 = q(x, y)dx − p(x, y)dy = 0. This equation defines two foliations with singularities, one on the real and one on the complex affine planes. We can compactify these foliations with singularities on the real respectively complex projective plane with homogeneous coordinates X, Y, Z . This is done as follows: Consider the pull-back of the form ω 1 via the map τ : We obtain a form r * (ω 1 ) = ω which has poles on Z = 0. Eliminating the denominators in the equationw = 0 we obtain an equation , C homogeneous polynomials of the same degree. The equation w = 0 defines a foliation with singularities on P 2 (K ) which, via the map (x,y)→ [x:y:1], extends the foliation with singularities, given by w 1 = 0 on K 2 to a foliation with singularities on P 2 (K ) which we call the compactification on the projective plane of the foliation with singularities defined by ω 1 = 0 on the affine plane K 2 . (K equal to R or C) This is because A, B, C are homogeneous polynomials over K , defined The points at infinity of the foliation defined by ω 1 = 0 on the affine plane are the singular points of the type [X : Y : 0] ∈ P 2 (K) and the line Z = 0 is called the line at infinity of this foliation. The singular points of the foliation F are the solutions of the three equations A = 0, B = 0, C = 0. In view of the definitions of A, B, C it is clear that the singular points at infinity are the points of intersection of Z = 0 with C = 0. For more details see [16], or [2] or [3]. An isolated singular point p at infinity of a polynomial vector field of degree n has two types of multiplicities: the maximum number m of finite singularities which can split from p, in small perturbations of the system within polynomial systems of degree n, and the maximum number m of infinite singularities which can split from p, in small such perturbations of the system. We encode the two in the column (m, m ) t . We then encode the global information about all isolated singularities at infinity using formal sums called cycles and divisors as defined in [19] or in [16] and used in [2,3,16,23]. We have two formal sums (divisors on the line at infinity Z = 0 of the complex affine plane) D S (P, Q; Z ) = w I w (P, Q)w and D S (C, Z ) = w I w (C, Z )w where w ∈ {Z = 0} and where by I w (F, G) we mean the intersection multiplicity at w of the curves F(X, Y, Z ) = 0 and G(X, Y, Z ) = 0 on the complex projective plane. For more details see [16]. Following [23] we encode the above two divisors on the line at infinity into just one but with values in the ring Z 2 : w.
For a system (1) with isolated finite singularities we consider the formal sum (zerocycle on the plane) D S ( p, q) = ω∈R 2 I w ( p, q)w encoding the multiplicities of all finite singularities. For more details see [1,16].

Some Geometrical Concepts
Firstly we recall some terminology.
We call elemental a singular point with its both eigenvalues not zero. We call semi-elemental a singular point with exactly one of its eigenvalues equal to zero. We call nilpotent a singular point with both its eigenvalues zero but with its Jacobian matrix at this point not identically zero. We call intricate a singular point with its Jacobian matrix identically zero.
The intricate singularities are usually called in the literature linearly zero. We use here the term intricate to indicate the rather complicated behavior of phase curves around such a singularity.
In this section we use the same concepts we considered in [2][3][4][5], such as orbit γ tangent to a semi-line L at p, well defined angle at p, characteristic orbit at a singular point p, characteristic angle at a singular point, characteristic direction at p. If a singular point has an infinite number of characteristic directions, we will call it a star-like point.
It is known that the neighborhood of any isolated singular point of a polynomial vector field, which is not a focus or a center, is formed by a finite number of sectors which could only be of three types: parabolic, hyperbolic and elliptic (see [13]). It is also known that any degenerate singular point can be desingularized by means of a finite number of changes of variables, called blow-up's, into elemental ans semielemental singular points (for more details see the Section on blow-up in [2] or [13]).
Topologically equivalent local phase portraits can be distinguished according to the algebraic properties of their phase curves. For example they can be distinguished algebraically in the case when the singularities possess distinct numbers of characteristic directions.
The usual definition of a sector is of topological nature and it is local, defined with respect to a neighborhood around the singular point. We work with a new notion, namely of geometric local sector, introduced in [2], based on the notion of borsec, term meaning "border of a sector" (a new kind of sector, i.e. geometric sector) which takes into account orbits tangent to the half-lines of the characteristic directions at a singular point. For example a generic or semi-elemental node p has two characteristic directions generating four half lines at p. For each one of these half lines at p there exists at least one orbit tangent to that half line at p and we pick an orbit tangent to that half line at p. Removing these four orbits together with the singular point, we are left with four sectors which we call geometric local sectors and we call borsecs these four orbits. The notion of geometric local sector and of borsec was extended for nilpotent and intricate singular points using the process of desingularization as indicated in [2]. We end up with the following definition: We call geometric local sector of a singular point p with respect to a sufficiently small neighborhood V , a region in V delimited by two consecutive borsecs. As mentioned these are defined using the desingularization process.
A nilpotent or intricate singular point can be desingularized by passing to polar coordinates or by using rational changes of coordinates. The first method has the inconvenience of using trigonometrical functions, and this becomes a serious problem when a chain of blow-ups are needed in order to complete the desingularization of the degenerate point. The second uses rational changes of coordinates, convenient for our polynomial systems. In such a case two blow-ups in different directions are needed and information from both must be glued together to obtain the desired portrait. Here for desingularization we use this second possibility, i.e. with rational changes of coordinates (x, y) → (x, zx) for the blow-up in the y-direction. This is a diffeomorphism for x = 0. The line x = 0 = y, the z-axis in R 3 , or x = 0 in the plane (x, z) is called the blow-up line. Analogously we use the change (x, y) → (zy, y) for the blow-up in the x-direction. This is a diffeomorphism for y = 0. The blow-up line is again the z-axis in R 3 or on the plane (y, z).
The two directional blow-ups can be reduced to only one 1-direction blow-up but making sure that the direction in which we do a blow-up is not a characteristic direction, not to lose information by blowing-up in the chosen direction. This can be easily solved by a simple linear change of coordinates of the type (x, y) → (x + ky, y) where k is a constant (usually 1). It seems natural to call this linear change a k-twist as the yaxis gets turned with some angle depending on k. It is obvious that the phase portrait of the degenerate point which is studied cannot depend on the set of k's used in the desingularization process.
We recall that after a complete desingularization all singular points are elemental or semi-elemental. For more details and a complete example of the desingularization of an intricate singular point see [4].
Generically a geometric local sector is defined by two borsecs arriving at the singular point with two different well defined angles and which are consecutive. If this sector is parabolic, then the solutions can arrive at the singular point with one of the two characteristic angles, and this is a geometrical information that can be revealed with the blow-up.
There is also the possibility that two borsecs defining a geometric local sector at a point p are tangent to the same halph-line at p. Such a sector will be called a cusp-like sector which can either be hyperbolic, elliptic or parabolic denoted by H , E and P respectively. In the case of parabolic sectors we want to include the information about how the orbits arrive at the singular points namely tangent to one or to the other borsec. We distinguish the two cases by writing P if they arrive tangent to the borsec limiting the previous sector in clockwise sense or P if they arrive tangent to the borsec limiting the next sector. In the case of a cusp-like parabolic sector, all orbits must arrive with only one well determined angle, but the distinction between P and P is still valid because it occurs at some stage of the desingularization and this can be algebraically determined. Example of descriptions of complicated intricate singular points are PE P HHH and E P H H P E.
A star-like point can either be a node or something much more complicated with elliptic and hyperbolic sectors included. In case there are hyperbolic sectors, they must be cusp-like. Elliptic sectors can either be cusp-like or star-like.

Notations for Singularities of Polynomial Differential Systems
In this work we limit ourselves to the class of quadratic systems with finite singularities of total multiplicity three and at most two real singularities. In [2] we introduced convenient notations which we also used in [3,4] some of which we also need here. Because these notations are essential for understanding the bifurcation diagram, we indicate below the notations needed for this article.
The finite singularities will be denoted by small letters and the infinite ones by capital letters. In a sequence of singular points we always place the finite ones first and then infinite ones, separating them by a semicolon';'.

Elemental Points:
We use the letters 's','S' for "saddles"; $ for "integrable saddles"; 'n', 'N ' for "nodes"; ' f ' for "foci"; 'c' (respectively c ; c ⊕ ) for "centers" (respectively "isochronous centers", "uniform isochronous centers") and © (respectively ©) for complex finite (respectively infinite) singularities. We distinguish the finite nodes as follows: • 'n' for a node with two distinct eigenvalues (generic node); • 'n d ' (a one direction node) for a node with two identical eigenvalues whose Jacobian matrix is not diagonal; • 'n * ' (a star node) for a node with two identical eigenvalues whose Jacobian matrix is diagonal. The case n d (and also n * ) corresponds to a real finite singular point with zero discriminant. The case of a finite complex singular point with zero discriminant (respectively trace) will be denoted by © τ (respectively by © ρ ).
In the case of an elemental infinite generic node, we want to distinguish whether the eigenvalue associated to the eigenvector directed towards the affine plane is, in absolute value, greater or lower than the eigenvalue associated to the eigenvector tangent to the line at infinity. This is relevant because this determines if all the orbits except one on the Poincaré disk arrive at infinity tangent to the line at infinity or transversal to this line. We will denote them as 'N ∞ ' and 'N f ' respectively.
Finite elemental foci and saddles are classified as strong or weak foci, respectively strong or weak saddles. The strong foci or saddles are those with non-zero trace of the Jacobian matrix evaluated at them. In this case we denote them by 's' and ' f '. When the trace is zero, except for centers, and saddles of infinite order (i.e. with all their Poincaré-Lyapounov constants equal to zero), it is known that the foci and saddles, in the quadratic case, may have up to 3 orders. We denote them by 's (i) ' and ' f (i) ' where i = 1, 2, 3 is the order. In addition we have the centers (respectively isochronous centers; uniform isochronous centers) which we denote by 'c' (respectively c ; c ⊕ ) and saddles of infinite order (integrable saddles) which we denote by '$'.
Foci and centers cannot appear as singular points at infinity and hence there is no need to introduce their order in this case. In case of saddles, we can have weak saddles at infinity but the maximum order of weak singularities in cubic systems is not yet known. For this reason, a complete study of weak saddles at infinity cannot be done at this stage. Due to this, in [2][3][4][5] and here we chose not even to distinguish between a saddle and a weak saddle at infinity.
All non-elemental singular points are multiple points, in the sense that there are perturbations which have at least two elemental singular points as close as we wish to the multiple point. For finite singular points we denote with a subindex their multiplicity as in 's (5) ' or in ' es (3) ' (the notation ' ' indicates that the saddle is semi-elemental and 's' indicates that the singular point is nilpotent, in this case a triple elliptic saddle (i.e. it has two sectors, one elliptic and one hyperbolic)). In order to describe the two kinds of multiplicity for infinite singular points we use the concepts and notations introduced in [23]. Thus we denote by ' a b ...' the maximum number a (respectively b) of finite (respectively infinite) singularities which can be obtained by perturbation of the multiple point. For example ' 1 1 S N ' means a saddle-node at infinity produced by the collision of one finite singularity with an infinite one; ' 0 3 S' means a saddle produced by the collision of 3 infinite singularities. The meaning of the notation ' ' in the general case will be described in the next paragraph.

Semi-elemental Points:
They can either be nodes, saddles or saddle-nodes, finite or infinite (see [13]). We denote the semi-elemental ones always with an overline, for example 'sn', 's' and 'n' with the corresponding multiplicity. In the case of infinite points we put ' ' on top of the parenthesis with multiplicities.
Moreover, in cases which will be later explained, an infinite saddle-node may be denoted by ' 1 1 N S' instead of ' 1 1 S N '. Semi-elemental nodes could never be 'n d ' or 'n * ' since their eigenvalues are always different. In case of an infinite semi-elemental node, the type of collision determines whether the point is denoted by 'N f ' or by . Nilpotent Points: They can either be saddles, nodes, saddle-nodes, elliptic saddles, cusps, foci or centers (see [13]). The first four of these could be at infinity. We denote the nilpotent singular points with a hat ' ' as in es (3) for a finite nilpotent elliptic saddle of multiplicity 3 and cp (2) for a finite nilpotent cusp point of multiplicity 2. In the case of nilpotent infinite points, we will put the ' ' on top of the parenthesis with multiplicity, for example 1 2 P E P − H (the meaning of P E P − H will be explained below). 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.
Intricate Points: It is known that the neighborhood of any singular point of a polynomial vector field (except for foci and centers) is formed by a finite number of sectors which could only be of three types: parabolic, hyperbolic and elliptic (see [13]). Then, a reasonable way to describe intricate and nilpotent points is to use a sequence formed by the types of their sectors. The description we give is the one which appears in the clockwise direction (starting anywhere) once the blow-down of the desingularization is done. Thus in non-degenerate quadratic systems (that is, the components of the system are coprime), we have just seven possibilities for finite intricate singular points of multiplicity four (see [6]) which are the following ones: phpphp (4) ; phph (4) ; hh (4) ; hhhhhh (4) ; peppep (4) ; pepe (4) ; ee (4) .
For infinite intricate and nilpotent singular points, we insert a dash (hyphen) between the sectors to split those which appear on one side or the other of the equator of the sphere. In this way we will distinguish between 2 2 P H P − P H P and 2 2 P P H − P P H. Whenever we have an infinite nilpotent or intricate singular point, we will always start with a sector bordering the infinity (to avoid using two dashes).
For the description of the topological phase portraits around the isolated singular points the information described above is sufficient. However we are interested in additional geometrical features such as the number of characteristic directions which figure in the final global picture of the desingularization. In order to add this information we need to introduce more notations. If two borsecs, limiting orbits of a sector, arrive at the singular point with the same direction, then the sector will be denoted by H , E or P . The index in this notation refers to the cusp-like form of limiting trajectories of the sectors. Moreover, in the case of parabolic sectors we need to distinguish whether the orbits arrive tangent to the half-line at the singular point of one borsec or to the other. We distinguish the two cases by P if they arrive tangent to the borsec limiting the previous sector in clockwise sense or P if they arrive tangent to the borsec limiting the next sector. A parabolic sector will be P * when all orbits arrive with all possible slopes between the two consecutive borsecs. In the case of a cusp-like parabolic sector, all orbits must be tangent to the same half-line at the singularity, but the distinction between P and P is still valid if we consider the different desingularizations we obtain from them.
Finally there is also the possibility that we have an infinite number of infinite singular points.

Line at Infinity Filled up with Singularities:
It is known that any such system has in a sufficiently small neighborhood of infinity one of 6 topological distinct phase portraits (see [24]). The way to determine these phase portraits is by studying the reduced systems on the infinite local charts after removing the degeneracy of the systems within these charts. In case a singular point still remains on the line at infinity we study such a point. In [24] the tangential behavior of the solution curves was not considered in the case of a node. If after the removal of the degeneracy in the local charts at infinity a node remains, this could either be of the type N d , N and N (this last case does not occur in quadratic systems as it was shown in [2]). Since no eigenvector of such a node N (for quadratic systems) will have the direction of the line at infinity we do not need to distinguish N f and N ∞ (see [2]). After removal of the degeneracy, other types of singular points at infinity of quadratic systems can be saddles, foci, centers, semi-elemental saddle-nodes or nilpotent elliptic saddles. We also have the possibility of no singularities after removal of the degeneracy. To convey the way these singularities were obtained as well as their nature, we use the notation

Affine Invariant Polynomials and Preliminary Results
Consider real quadratic systems of the form with homogeneous polynomials p i and q i (i = 0, 1, 2) in x, y which are defined as follows: It is known that on the set QS of all quadratic differential systems (2) acts the group Aff(2, R) of affine transformations on the plane (cf. [23]). For every subgroup G ⊆ Aff(2, R) we have an induced action of G on QS. We can identify the set QS of systems (2) with a subset of R 12 via the map QS−→ R 12 which associates to each system (2) the 12-tupleã = (a 00 , . . . , b 02 ) of its coefficients. We associate to this group action polynomials in x, y and parameters which behave well with respect to this action, the G L-comitants, the T -comitants and the CT -comitants. For their constructions we refer the reader to the paper [23] (see also [25]). In the statement of our main theorem intervene the following 22 invariant polynomials constructed in these article: In the proof of our main theorem there appear 19 other invariant polynomials: We refer the reader interested in the construction of these polynomials to the following associated web-page: http://mat.uab.es/~artes/articles/qvfinvariants/qvfinvariants.html. We need here the following results concerning with isochronous and uniformly isochronous centers (see [10,12,17]). (2) has an isochronous center on its phase plane if and only if this system can be brought via an affine change of coordinates and a rescaling of the time to one of the following four systems:

Lemma 2 ([12]) Assume that a system (1) has a center at the origin of coordinates. Then, this center is uniformly isochronous if and only if due to a linear change of variables and a rescaling of time this system can be written under the form
where R(x, y) is a polynomial in x and y of degree m − 1, and R(0, 0) = 0.
Remark 2 Considering Lemma 2 we observe that only the system (S 2 ) is of the form (3) (with R(x, y) = x) and hence it is uniformly isochronous.

The Proof of the Main Theorem
According to [26] for a quadratic system to have finite singularities of total multiplicity three (i.e. m f = 3) the conditions μ 0 = 0 and μ 1 = 0 must be satisfied. Since the subclass with three real finite simple singularities was considered in [5] (i.e. the additional condition D < 0 holds) we consider here three subclasses: • systems with one real and two complex finite singularities (μ 0 = 0, μ 1 = 0, D > 0); • systems with one double and one simple real finite singularities (μ 0 = 0, μ 1 = 0, We observe that the systems from each one of the above mentioned subclasses have finite singularities of total multiplicity 3 and therefore by [2] the following lemma is valid.

Lemma 3
The geometric configurations of singularities at infinity of the family of quadratic systems possessing finite singularities of total multiplicity 3 (i.e. μ 0 = 0, μ 1 = 0) are classified in Fig. 4 according to the geometric equivalence relation.

Systems With One Real and Two Complex Finite Singularities
Consider quadratic systems (2) with real coefficients and variables x and y. Assume that these systems possess one real and two complex finite singularities. Then according to [26] via an affine transformation and time rescaling these systems could be brought to the form: possessing the real elemental singular point M 1 (0, 0) and the two complex singularities M 2,3 (u ± i, 1). For these systems calculations yield Since for the above systems we have μ 1 = 0 (i.e. gm − hl = 0) we observe that the condition κ = 0 is equivalent to h = 0.

The Case κ = 0
Then h = 0 and we may assume h = 1 due to a time rescaling. So we get the systemṡ and for the singular points M 1 (0, 0) and M 2,3 (u ± i, 1) we have the following values for the traces ρ i , for the determinants i , for the discriminants τ i and for the linearization matrix M 1 (in the case of a real singularity): Then for the above mentioned systems we calculate Remark 3 We observe that M = 0 and the condition κ = 0 implies K G 9 = 0. Moreover we have sign ( K ) = −sign (κ).
1) Assume first T 4 = 0. Then the focus is strong and considering Lemma 3 and the condition M = 0 (see Remark 3) we arrive at the following three configurations of singularities: a) The case T 3 = 0. Considering (7) we have ρ 1 = 0 (i.e. l = 2(gu − 1)/(1 + u 2 )) and we calculate a 1 ) The subcase F 1 = 0. As T 3 F < 0 considering [26] (see Main Theorem, (b 1 )) we have a first order weak focus. So by Lemma 3 and Remark 3 we get the three global configurations: a 2 ) The subcase F 1 = 0. Then the order of the weak focus is at least two. We claim that in this case the condition gu − 1 = 0 holds. Indeed if gu − 1 = 0 (then u = 0) we set g = 1/u (this implies l = 2(gu − 1)/(1 + u 2 ) = 0) and we calculate for systems (5) Therefore due to κ = 0 we have F 1 = 0 and we obtain a contradiction which proves our claim. So gu − 1 = 0 and considering (8) the condition F 1 = 0 gives and calculations yield where is a quadratic polynomial with respect to g whose discriminant equals −16u 4 (3 + 2u 2 ) ≤ 0. So for the expressions above we get η > 0 and we observe that the conditions F 2 = 0 and T 3 = 0 imply 5gu − 4 = 0. In this case u = 0, otherwise we have F 2 = 0. Therefore we get g = 4/(5u) and we calculate: Hence by [26] we could not have a center. So considering Lemma 3 we arrive at the two global configurations: (7) it is clear that the condition T 4 = T 3 = 0 gives ρ 2 = ρ 3 = 0 and this implies g = m and l = 2mu/(1 + u 2 ). Then calculations yield We claim that in this case the condition W 4 < 0 implies η < 0. Indeed, we observe that the curves η = 0 and τ 1 = 0 have only complex points of intersection. So it is sufficient to determine the sign of the polynomial η at a point where τ 1 < 0. At the point (m, u) = (−1/2, −2) we have τ 1 = −2 < 0 and η = −19/5 < 0 and this proves our claim.
Indeed taking into account the linearization matrix M 1 from (6) it is clear that we have a star node if and only if g = 0, l = 2/(1 + u 2 ) and m = 2u/(1 + u 2 ). In this case a straightforward computation gives us U 3 = 0.
Conversely, assume that U 3 = 0. Then we have and this implies So due to κ = 0 we obtain l = 2/(1 + u 2 ) and then we have Therefore we get m = 2u/(1 + u 2 ) and this proves our claim. We observe that in the case U 3 = 0 we obtain η = −64/(1 + u 2 ) 2 < 0 and according to Lemma 3 we get the three global configurations of singularities 2) Suppose now W 3 = 0. Considering (7) we get τ 2 = τ 3 = 0. However for systems (5) these conditions lead to a quite complicated system of polynomial equations. So we decide to construct a more convenient canonical form and this will be done in Lemma 4. According to this lemma in the case κ < 0 we obtain 12 geometrically distinct configurations and the corresponding invariant conditions which characterize them.
The subcase F 1 = 0. Then we have a weak saddle whose order is at least two. We claim that in this case the condition F 2 = 0 holds, i.e. we could not have a third order weak saddle. Indeed, as it was shown earlier (see the compartment 3.1.1.1.1, p. a 2 )) the conditions T 4 = F 1 = 0 yields and then we obtain the value of F 2 given in (9). So due to T 3 = 0 the condition F 2 = 0 implies 5gu − 4 = 0. In this case u = 0, otherwise we have F 2 = 0. Therefore we get g = 4/(5u) and we calculate: and we obtain a contradiction which proves our claim. Thus we obtain the unique global configurations of singularities which is: (7) it is clear that the condition T 4 = T 3 = 0 gives ρ 2 = ρ 3 = 0 and this implies g = m and l = 2mu/(1 + u 2 ). Then we have ρ 1 = 2 = 0, i.e. the saddle is strong: The Possibility W 4 = 0. Since the real singularity is a saddle we conclude that in this case both complex points have zero discriminants (i.e. W 4 = W 3 = 0). In this case by Lemma 4 we get the three geometrically distinct configurations (C f 13 ), (C f 14 ) and (C f 15 ) and the corresponding invariant conditions which characterize them.
a) The case B 1 = 0. Then ρ 1 = 0 and the focus is strong. Considering Remarks 4 and 5 we arrive at the next six global configurations of singularities we obtain ρ 2 ρ 3 = 0 due to K > 0 (i.e. m < 0). So we get ρ 1 = 0 and this gives l = 2u/(u 2 + 1). Then following [26] we calculate and clearly the condition K > 0 implies B 2 < 0 and σ = 0. So according to [26] (see Main Theorem) we have a weak focus of the first order if F 1 = 0 (the statement (e 2 )) and we have a center if F 1 = 0 (the statement (e 4 ), [β]). b 1 ) The subcase F 1 = 0. Then u = 0 (this implies C 2 = 0) and we arrive at the next five global configurations of singularities (1) , ©, ©; 1 3 H H P − P : Example ⇒ (l = 1, m = −1/2, u = 1) (ifL = 0). b 2 ) The subcase F 1 = 0. Then u = 0 and hence l = 0 and this leads to the one-parameter family of systems: This family of systems possesses for m < 0 a center and two complex singularities. We claim that the system (S 2 ) as well as the system (S 3 ) (see Lemma 1) belong to this family for some specific values of the parameter m. Indeed, it is easy to observe that for m = −1/2, after the rescaling (x, y, t) → (−x, −y, −t) we get exactly the system (S 2 ). Considering Remark 2 we conclude that the system (16) corresponding to m = −1/2 possesses a uniformly isochronous one. Considering (14) we deduce that the condition m = −1/2 is equivalent toL = 0.

Evidently the system (S 3 ) belongs to this family if and only if
On the other hand for systems (16) we calculate I = 3m(1 − m)(2 + m) and hence due to m < 0 the condition m = −2 is equivalent to I = 0.
Considering (14) we conclude that in this case the conditionL = 0 implies also C 2 = 0 and in the case I = 0 (i.e. m = −2) we have N < 0. So according to Lemma 3 we arrive at the next 6 configurations of singularities: 2) Suppose now W 7 > 0. In this case we have τ 1 > 0 and the anti-saddle is a generic node.
By this remark we conclude that the condition W 7 > 0 implies C 2 = 0. So considering Remarks 4 and 5 we arrive at the next five global configurations of singularities a) The case W 6 = 0. Then τ 2 τ 3 = 0 and we get τ 1 = 0, i.e. the real singularity is a node with coinciding eigenvalues. Considering the corresponding linearization matrix M 1 given by (12) we conclude that this node could not be a star node. Thus considering Remarks 4, 5 and 6 we arrive at the next five global configurations of singularities: The case W 6 = 0. By (13) the conditions W 7 = W 6 = 0 give τ 2 = τ 3 = 0 and this implies for systems (11) As K > 0 (i.e. m < 0) we have m − 1 = 0 and hence the equations above yield Then we calculate: The subcase W 5 < 0. Then τ 1 < 0 and the real singular point is a focus. Clearly the focus is strong if ρ 1 = 0 (i.e. B 1 = 0) and it is weak if ρ 1 = 0. In the second case we have m = −1 and then calculations yield According to [26] (see Main Theorem, the statement (e 4 ), [β]) we have a center. On the other hand, due to K = 0, the condition B 1 = 0 is equivalent to N = 0. So considering Remarks 4, 5 and 6 we arrive at the next five global configurations of singularities we arrive at the two configurations Thus the case κ = 0 is completely examined.
In what follows we consider the family of systems (5) with the special property: the discriminants corresponding to the complex points vanish. However for these systems the discriminants are polynomials quite big which imply complicated computations. In order to avoid such computations we will use a new normal form of this family of systems.
On the other hand the invariant polynomial μ 0 is the discriminant of the form K (ã, x, y). So in this case the homogeneous quadratic polynomialK has the form: K = (ux + vy) 2 . Moreover the linear form ux + vy is a common factor of the quadratic parts of systems (2) and, hence, the fourth finite singularity has coalesced with the infinite singular point N (−v, u, 0). We observe that the condition v = 0 has to be satisfied, otherwise the common factor of the quadratic parts of systems (18) will be x and this implies a = b = 0 and we get degenerate systems. So v = 0 and via the transformation x 1 = x and y 1 = ux/v + y, which preserves the singular points M 1,2 (0, ±i), we getK (ã, x, y) = δy 2 . This means that the common factor of the homogeneous quadratic parts of systems (18) will be y and therefore the transformation applied, implies the conditions g = l = 0.
Thus we arrive at the family of systemṡ possessing three distinct finite singularities: M 1,2 (0, ±i) and one real singularity M 3 .
A. The Possibility a = 0 Then we may assume a = 1 due to a time rescaling and we calculate e = c(b − h)/2, m = c 2 − 4(b − h) 2 /16. So we get the family of systemṡ possessing the complex singular points M 1,2 (0, ±i) and one real elemental singularity M 3 . For these systems we calculate: 32 , where

The Case κ < 0
ThenK > 0 and according to [6] (see the Table 1, lines 124-135) the elemental real singular point is an anti-saddle and its type is governed by the invariant polynomial W 2 , as in the case under consideration we have G 9 = 0 and W 4 = W 3 = 0.
On the other hand according to [1] we have the next result.

Remark 7
In the class of quadratic systems with a weak focus of order two the condition μ 0 = 0 = W 4 implies the existence of exactly three distinct real singularities.
So by this remark, systems (20) could not have a weak focus of the second order. Therefore in the case considered, we arrive at the following three configurations: ;

The
Subcase W 2 > 0. Then τ 3 > 0, i.e. the real singular point is a generic node. In this case we get the next three configurations: 3.1.2.3. The Subcase W 2 = 0. Then τ 3 = 0 and the real singularity is a node with coinciding eigenvalues. We claim that we could not have a star node in this case. Indeed, assume that the singular point M 3 is a star node. According to [3, Lemma 5] the conditions either (i) or (ii) of this lemma must hold. Suppose first that the conditions (i) are satisfied, i.e. U 1 = 0, U 2 = 0, U 3 = Y 1 = 0. Then we must have We observe that the condition c = 0 implies U 2 = 0. So, c = 0 and then clearly the condition (b − 3h)(b + h) ≥ 0 must hold. Obviously we could assume that both factors are non-negative and we set new parameters u and v as follows: As the third factor can be obtained by the second one replacing v by −v we may assume cu − u 2 v − v 3 = 0. If u = 0 then v = 0 and we obtain U 3 = −27c 4 y(c 2 x 2 + 16y 2 ) 2 /2 12 .
Since c = 0 we get U 3 = 0, i.e. the conditions (i) of the lemma are not satisfied.
Thus we get a node n d and we obtain the three configurations:

The Case κ > 0
ThenK < 0 and according to [6] (see the Table 1, line 123) the elemental real singular point is a saddle. In this case the total index of the infinite singularities must be +2 and hence, at infinity besides a saddle-node we have two nodes (i.e. η > 0).

3.1.3.2.2.
The possibility F 1 = 0. In this case we have a weak saddle of order two and it could not be of order three. Indeed as it follows from [16] the condition μ 0 = 0 implies D < 0, i.e. we could not have one real and two complex singular points. So we arrive at the configuration B. The Possibility a = 0 In this case we for systems (19) we get τ 1,2 = −(2b − 2h + ic) 2 = 0 that implies c = 0 and h = b. Hence we get the family of systemṡ for which we have μ 1 = −4eb 3 y = 0 and therefore we can consider e = 1 = b due to the rescaling (x, y, t) → (bx/e, y, t/b) and then we calculate Since κ = 0 we get κ < 0, η > 0, T 4 = 0 and sign (W 2 ) = sign (m 2 − 2). So we could only have the configurations (C f 2 ), (C f 8 ) and (C f 11 ), which are already detected in the case a = 0.
As all the cases are considered, the lemma is proved.

Systems with One Double and One Simple Real Finite Singularities
Assume that quadratic systems (2) possess one double and one simple finite singularities. Then according to [26] via an affine transformation and time rescaling these systems could be brought to the form: x = cx + cy − cx 2 + 2hx y,ẏ = ex + ey − ex 2 + 2mx y (22) possessing the double singular point M 1,2 (0, 0) and the elemental singularity M 3 (1, 0). For these systems calculations yield As for systems above we have μ 1 = 0 (i.e. gm −hl = 0) we observe that the condition κ = 0 is equivalent to h = 0.

The Case κ = 0
Then h = 0 and we may assume h = 1 due to a time rescaling. So we get the systemṡ possessing the singular points M 1,2 (0, 0) and M 3 (1, 0). For these singularities we have the following values of the traces ρ i , of the determinants i and of the discriminants τ i : Then for the above systems we have Remark 8 We observe that M = 0 and the condition κ = 0 implies K G 9 = 0. Moreover we have sign ( K ) = −sign (κ).
3.2.1.1. The Subcase κ < 0 This implies K > 0 and according to [6] (see Table 1, lines 138-144) the elemental singular point is an anti-saddle and its type is governed by the invariant polynomials W 4 and W 2 (as in the considered case we have G 9 = 0). 3.2.1.1.1. The possibility W 4 < 0. Considering (24) we have τ 3 < 0, i.e. the elemental singular point is a focus. On the other hand as the condition W 4 = 0 implies E 1 = 0 and G 9 = 0, according to [6] (see Table 1, line 140) the double point is a semi-elemental saddle-node. 1) Assume first T 4 = 0. Then the focus is strong and considering Lemma 3 and M = 0 from (24) we arrive at the following three configurations of singularities: • f, sn (2) (2) ; 0 2 S N, 1 1 S N : Example ⇒ (c = 2, e = 1/2, m = 0) (if η = 0). 2) Suppose now T 4 = 0. Then considering (24) we get ρ 3 = 0, i.e. the focus is weak. In this case we have e = c − 2m and we calculate So the condition W 4 < 0 gives T 3 F < 0 and by [26] the order of weak focus is governed by invariant polynomial F 1 . a) The case F 1 = 0. Then the simple singular point of systems (23) is a first order weak focus and considering Lemma 3 we arrive at the following three global configurations of singularities: • f (1) , sn (2) (1) , sn (2) 1 + c)). Then we calculate 5 (1 + c) 9 , κ = 64 and we observe that due to κ = 0 the condition η > 0 holds. Moreover, we get F 2 = 0 for c = −6/5 and in this case the weak focus is of order 3 as F 3 F 4 = 2 23 3 21 5 −13 = 0 (see [26], the Main Theorem). Thus considering Lemma 3 we get the following two global configurations of singularities: • f (2) , sn (2)  On the other hand when we have a star node we obtain η = 0. So considering Lemma 3 we arrive at the following four global configurations of singularities: • n d , sn (2) . 2) Suppose now T 4 = 0. As it was mention above in this case we have ρ 1 = 0 and then E 1 = 0. As K > 0 and G 9 = 0 according to [6] (see Table 1, lines 142 and 144) the double point is a nilpotent cusp, whereas the type of the anti-saddle in this case is governed by the invariant polynomial W 2 . Setting e = −c for the elemental singular point we have and calculations yield a) The case W 2 < 0. This gives τ 3 < 0 and the elemental singularity is a focus. a 1 ) The subcase T 2 = 0. Then ρ 3 = 0 and the focus is strong. So considering Lemma 3 we arrive at the next three global configurations of singularities: • f, cp (2) So the condition κ = 0 yields T 1 F 1 = 0 (moreover κ < 0 implies H < 0) and according to [26] (see the statement (d 2 ) of Main Theorem) the weak focus could only be of order one. So we get the next three global configurations of singularities: (1) , cp (2)  c) The case W 2 = 0. Considering (27) we have τ 3 = 0 and systems (23) with e = −c possess a node with coinciding eigenvalues. As the linearization matrix has the form given in (26) and c = 0 (due to κ = 0) this node could not be a star node. So considering Lemma 3 we arrive at the following three global configurations of singularities: • n d , cp (2)

The
Subcase κ > 0 This implies K < 0 and according to [6] (see Table 1, lines 136, 137) the elemental singular point is a saddle. At the same time the type of the double point depends on the value of the invariant polynomial E 1 .
On the other hand considering (24) we observe that the condition κ > 0 implies η > 0 and according to Lemma 3 at infinity there could be the unique configuration of singularities: 1 1 S N, N f , N f . 3.2.1.2.1. The Possibility T 4 = 0. By (24) this implies ρ 1 ρ 3 = 0 and then E 1 = 0, i.e. the double point is semi-elemental saddle-node. On the other hand as ρ 3 = 0 the saddle is strong and we obtain the following global configuration of singularities: • s, sn (2)  1) Assume first T 3 = 0. In this case we get ρ 1 = 0 (then E 1 = 0) and ρ 3 = 0. We get e = c − 2m and by (25) and [26] we obtain a weak saddle the order of which is governed by the invariant polynomial F 1 .
a) The case T 2 = 0. Considering (27) we get ρ 3 = 0 and therefore we have a strong saddle: • s, cp (2) The case T 2 = 0. Then m = c and taking into account (28) we conclude, that the weak saddle could be only of the first order: • s (1) , cp (2)
In this case h = 0 and then c = 0, otherwise systems (22) become degenerate. So we may assume c = 1 due to a time rescaling and we obtain the 2-parameter family of systems for the singular points M 1,2 (0, 0) and M 3 (1, 0) of which we have the following linearization matrices, traces, determinants and discriminants: For the systems above calculations yield We observe that the condition μ 1 = 0 implies K = 0. According to [6] and [26] the polynomials above are responsible for the types of the finite singularities of systems (29). In order to describe the infinite singularities considering Lemma 3 we calculate the additional invariant polynomials: So considering Lemma 3 we have the next remark.
Remark 9 Systems (29) could possess at infinity only one of the following seven configurations of singularities: 3.2.2.1. The Subcase K < 0 Then 3 < 0 and the elemental singularity is a saddle. 3.2.2.1.1. The Possibility B 1 = 0. In this case we have ρ 1 ρ 3 = 0 and therefore the double point is a semi-elemental saddle-node and the saddle is strong. We observe that the condition K < 0 implies m > 0 and then M = 0. So considering the above remark we arrive at the global configuration of singularities • s, sn (2) ; 1 2 P E P − H, N f : Example ⇒ (e = 0, m = 1).

3.2.2.2.
The subcase K > 0 In this case according to [6] (see Table 1, lines 139-143) the elemental singular point is an anti-saddle and its type is governed by the invariant polynomial W 7 (as in the considered case we have G 9 = 0).
As for the above systems we have μ 1 = 0 (i.e. gm − hl = 0) we observe that the condition κ = 0 is equivalent to h = 0.
3.3.1.1. The Subcase κ < 0 This implies K > 0 and according to to [6] (see Table  1, lines 146, 147) the triple finite singular point is a semi-elemental node if G 10 = 0 and it is a nilpotent elliptic saddle if G 10 = 0 (in this case l = 0 due to κ = 0). As M = 0 considering Lemma 3 we arrive at the following five configurations of singularities:
Thus considering Lemma 3 we obtain the following two configurations of singularities:

The Case κ = 0
Then h = 0 and gm = 0 due to μ 1 = 4g 2 m 2 x = 0. So we may assume g = 1 due to the rescaling (x, y) → (x/g, y/g 2 ) and we get the family of systemṡ x = y + x 2 ,ẏ = ly + lx 2 + 2mx y, m = 0, for which we calculate μ 0 = κ = η = 0, μ 1 = 4m 2 x, K = 4mx 2 , G 10 = l 3 m 3 , 3.3.2.1. The Subcase K < 0 According to to [6] (see Table 1, line 145) the triple finite singular point is saddle. As it was mentioned earlier this triple saddle is semielemental if G 10 = 0 and it is a nilpotent saddle if G 10 = 0, which is equivalent to F 1 = 0. On the other hand the condition K < 0 implies m < 0 and then M = 0. Therefore considering Lemma 3 we obtain the following two configurations of singularities: Thus we obtained that a quadratic system with a triple finite singularity possesses only one of the 19 global configurations of singularities given above. As all the cases are examined, we have constructed all 155 possible configurations for the family of quadratic systems with m f = 3 possessing at most two real finite singularities. Therefore our Main Theorem is completely proved.