Structurally Unstable Quadratic Vector Fields of Codimension Two: Families Possessing Either a Cusp Point or Two Finite Saddle-Nodes

The goal of this paper is to contribute to the classification of the phase portraits of planar quadratic differential systems according to their structural stability. Artés et al. (Mem Am Math Soc 134:639, 1998) proved that there exist 44 structurally stable topologically distinct phase portraits in the Poincaré disc modulo limit cycles in this family, and Artés et al. (Structurally unstable quadratic vector fields of codimension one, Springer, Berlin, 2018) showed the existence of at least 204 (at most 211) structurally unstable topologically distinct phase portraits of codimension-one quadratic systems, modulo limit cycles. In this work we begin the classification of planar quadratic systems of codimension two in the structural stability. Combining the sets of codimension-one quadratic vector fields one to each other, we obtain ten new sets. Here we consider set AA obtained by the coalescence of two finite singular points, yielding either a triple saddle, or a triple node, or a cusp point, or two saddle-nodes. We obtain all the possible topological phase portraits of set AA and prove their realization. We got 34 new topologically distinct phase portraits in the Poincaré disc modulo limit cycles. Moreover, in this paper we correct a mistake made by the authors in the book of Artés et al. (Structurally unstable quadratic vector fields of codimension one, Springer, Berlin, 2018) and we reduce to 203 the number of topologically distinct phase portrait of codimension one modulo limit cycles.


Introduction
Mathematicians are fascinated in closing problems. Having a question solved or even sign with a "q.e.d" a question asked in the past is a pleasure which is directly proportional to the time elapsed between the formulation of the question and the moment of the answer.
With the advent of the differential calculus, it opened the possibility of solving many questions that medieval mathematicians asked, but at the same time it made the field of questions formulated even bigger. The search for primitive functions that could not be expressed algebraically or with a finite number of analytic terms complicated the future research lines, and even new areas of mathematics were created to give answers to these questions. And besides the problem of finding a primitive to a differential equation in a single dimension, if we add the possibility of more dimensions, the problem becomes much more difficult.
Therefore, it took almost 200 years between the approach of the first linear differential equations and its complete resolution by Laplace in 1812. After the resolution of linear differential systems, for any dimension, it seemed natural to address the classification of quadratic differential systems. However, it was found that the problem would not have an easy and fast solution. Unlike the linear systems that can be solved analytically, quadratic systems (not even, therefore, those of higher degree) generically admit a solution of that kind, at least, with a finite number of terms.
Therefore, for the resolution of non-linear differential systems, another strategy was chosen and it allowed the creation of a new area of knowledge in Mathematics: the Qualitative Theory of Differential Equations [24]. Since we are not able to give a concrete mathematical expression to the solution of a system of differential equations, this theory intends to express by means of a complete and precise drawing the behavior of any particle located in a vector field governed by such a differential equation, i.e. its phase portrait.
Even with all the reductions made to the problem until now, there are still difficulties. The most expressive difficulty is that the phase portraits of differential systems may have invariant sets that are not punctual, as the limit cycles. A linear system cannot generate limit cycles; at most they can present a completely circular phase portrait where all the orbits are periodic. But a differential system in the plane, polynomial or not, and starting with the quadratic ones, may present several of these limit cycles. It is trivial to verify that there can be an infinite number of these cycles in non-polynomial problems, but the intuition seems to indicate that a polynomial system should not have an infinite number of limit cycle since it cannot have an infinite number of isolated singular points. And because the number of singular points is linked to the degree of the polynomial system, it also seems logical to think that the number of limit cycles could also have a similar link, either directly as the number of singular points, or even in an indirect way from the number the parameters of such systems.
In 1900, Hilbert [16,17] proposed a set of 23 problems to be solved in the twentieth century, and among them his well-known 16th problem asks for the maximum number of limit cycles H (n) a polynomial differential system in the plane with degree n may have. More than one hundred years after, we do not have an uniform upper bound for this generic problem, only for specific families of such a system.
In 1966, Coppel [11] claimed to believe that the classification of quadratic systems should be able to be completed in purely algebraic terms. That is, by means of algebraic equalities and inequalities, it should be possible to determine the phase portrait of a quadratic system.
His proclamation was not easy to refute at that time, since the isolated finite singular points of a quadratic system can be found by means of the resultant that is of fourth degree, and its solutions can be calculated algebraically, like those of infinity. Moreover, it was known at that time to generate cycles limits by a Hopf bifurcation, whose conditions are also determined algebraically.
On the other hand, in 1991, Dumortier and Fiddelears [12] showed that, starting with the quadratic systems (and following all the higher-degree systems), there exist geometric and topological phenomena in phase portraits of such a system whose determination cannot be fixed by means of algebraic expressions. More specifically, most of the connections among separatrices and the occurrence of double or semi-stable limit cycles are not algebraically determinable.
Therefore, the complete classification of quadratic systems is a very difficult task at the moment and it depends enormously of the culmination of Hilbert's 16th problem, even at least partially for H (2).
Even so, a lot of problems have been appearing related to quadratic systems and to which it has been possible to give an answer. In fact, there are more than one thousand articles published directly related to quadratic systems. John Reyn, from Delft University (Netherlands), was committed in preparing bibliography that was published several times until his retirement (see [25,[27][28][29][30]). It is worth mentioning that in the last two decades many other articles related to quadratic systems have appeared, what figures that the mentioned amount of one thousand papers in that bibliography has already been widely exceeded.
Many of the questions proposed and the problems solved have dealt with subclassifications of quadratic systems, that is, classifications of systems that shared some characteristic in common. For instance, we have systems with a center [32,33], with a weak focus of third order [4,20], with a nilpotent singularity [19], without real singular points [14], with two invariant lines [25] and so on, up to a thousand articles. In some of them complete answers could be given, including the problem of limit cycles (the existence and the number of limit cycles), but in other cases, the classification was done modulo limit cycles, that is, all the possible phase portraits without taking into account the presence and number of cycles. Since in quadratic systems a limit cycle can only surround a single finite singular point, and which must necessarily be a focus [11], then it is enough to identify the outermost limit cycle of a nesting of cycles with a point, and interpret the stability of that point as the outer stability of this cycle, and study everything that can happen to the phase portrait in the rest of the space.
Within the families of quadratic systems that were studied in the twentieth century, we would highlight the study of the structurally stable quadratic systems, modulo limit cycles. That is, the goal was to determine how many and which phase portraits of a quadratic system cannot be modified by small perturbations in their coefficients. To obtain a structurally stable system modulo limits cycles we need few conditions: we do not allow the existence of multiple singular points and the existence of connections of separatrices. Centers, weak foci, semi-stable cycles, and all other unstable elements are submerged in the quotient modulo limit cycles. This systematic analysis [3] showed that the structurally stable quadratic systems sum a total of 44 topologically distinct phase portraits.
Once assumed that, if we get to obtain a global classification of quadratic systems before solving Hilbert's 16th problem, this will have to be modulo limit cycles. We proposed to carry out a systematic global classification and, for this, we cannot be attained only to the study of families of systems that do not give more than extremely local visions of global parameter space. Even applying to our quadratic system a linear change of coordinates plus a translation and a time rescaling, which supposes a reduction from the initial 12 parameters to a limited set of systems with 5 parameters, R 5 is still a very large space.
There are two ways to carry out a systematic study of all the phase portraits of the quadratic systems. One of them is the one initiated by Reyn in which he studied the phase portraits of all the quadratic systems in which all the finite singular points have coalesced with infinite singular points [26]. Later, he studied those in which exactly three finite singular points have coalesced with points of infinity, so there remains one finite and real. And then he completed the study of the cases in which two finite singular points have coalesced with points of infinity, originating two real points, or one double point, or two complex points. His work on finite multiplicity three was incomplete and finite multiplicity four was unaffordable.
The other approach, instead of working from the highest degrees of degeneracy to the lower ones, is going in the contrary direction. We already know that the structurally stable quadratic systems sum 44 topologically distinct phase portrait, as we mentioned above. The natural problem to be studied after was the structurally unstable quadratic differential systems of codimension one. This study [5] was done in approximately 20 years and finally we obtained at least 204 (and at most 211) topologically phase portraits of codimension one modulo limit cycles.
The next step is to study the structurally unstable quadratic systems of codimension two, modulo limit cycles. The approach is the same used in the previous two works [3,5]. We start looking for all the topologically possible phase portraits of codimension two, and then try to realize all of them or show that some of them are impossible.
Since there are more than 10 cases of codimension two to be analyzed, it should be impracticable to perform a single paper with all the results. So we decided to split it in several papers, and this present article is the first one of this series.
In what follows, we recall some definition and notation used in this paper, and then we explain all these cases of structurally unstable quadratic systems of codimension two, one by one, and present the completion of the first case.
Let X be a vector field. A point p ∈ R 2 such that X ( p) = 0 (respectively X ( p) = 0) is called a singular point (respectively regular point) of the vector field X .
Let P n (R 2 ) be the set of all polynomial vector fields on R 2 of the form X (x, y) = (P(x, y), Q(x, y)), with P and Q polynomials in the variables x and y of degree at most n (with n ∈ N). In this set we consider the coefficient topology by identifying each vector field X ∈ P n (R 2 ) with a point of R (n+1)(n+2) (see more details in [5]).
For X ∈ P n (R 2 ), we consider the Poincaré compactified vector field p(X ) corresponding to X as the vector field induced on S 2 as described in [1,5,13,15,31]. Concerning this, a singular point q of X ∈ P n (R 2 ) is called infinite (respectively finite) if it is a singular point of p(X ) in S 1 (respectively in S 2 \ S 1 ). Now, we present the local classification of the singular points of p(X ). Let q be a singular point of p(X ).
The classical definitions are: • q is non-degenerate if det (Dp(X )(q)) = 0, i.e. the determinant of the linear part of p(X ) at the singular point q is nonzero; • q is hyperbolic if the two eigenvalues of Dp(X )(q) have real part different from 0; • q is semi-hyperbolic if exactly one eigenvalue of Dp(X )(q) is equal to 0.
However, we also may use new notation introduced in [6] directly related to the Jacobian matrix of the singularity. We have: • q is elemental if both of its eigenvalues are non-zero; • q is semi-elemental if exactly one of its eigenvalues equals to zero; • q is nilpotent if both of its eigenvalues are zero, but its Jacobian matrix at this point is non-identically zero; • q is intricate if its Jacobian matrix is identically zero; • q is an elemental saddle if det (Dp(X )(q)) < 0, i.e. the product of the eigenvalues of Dp(X )(q) is negative; • q is an elemental antisaddle if det (Dp(X )(q)) > 0 and the neighborhood of q is not formed by periodic orbits, in which case we would call it a center, i.e., it is either a node or a focus.
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.

Remark 1
Saddles have always (topological) index −1 and antisaddles have index +1 (see [13,18] for the definition of index of a singular point).
We encourage the reader to recall the definition of characteristic directions and finite sectorial decomposition of vector fields p(X ) ∈ P n (S 2 ) (or X ∈ P n (R 2 )) (for instance, see [13]).
Let p(X ) ∈ P n (S 2 ) (respectively X ∈ P n (R 2 )). A separatrix of p(X ) (respectively X ) is an orbit which is either a singular point (respectively a finite singular point), or a limit cycle, or a trajectory which lies in the boundary of a hyperbolic sector at a singular point (respectively a finite singular point). Neumann [21] proved that the set formed by all separatrices of p(X ), denoted by S( p(X )), is closed. The open connected components of S 2 \ S( p(X )) are called canonical regions of p(X ). We define a separatrix configuration as the union of S( p(X )) plus one representative solution chosen from each canonical region. Two separatrix configurations S 1 and S 2 of vector fields of P n (S 2 ) (respectively P n (R 2 )) are said to be topologically equivalent if there is an orientation-preserving homeomorphism of S 2 (respectively R 2 ) which maps the trajectories of S 1 onto the trajectories of S 2 .
We define skeleton of separatrices as the union of S( p(X )) without the representative solution of each canonical region. Thus, a skeleton of separatrices can still produce different separatrix configurations.
In this paper we call a heteroclinic orbit as a separatrix which starts and ends on different points and a homoclinic orbit as a separatrix which starts and ends at the same point. A loop is formed by a homoclinic orbit and its associated singular point. These orbits are also called separatrix connections.
A vector field p(X ) ∈ P n (S 2 ) is said to be structurally stable with respect to perturbations in P n (S 2 ) if there exists a neighborhood V of p(X ) in P n (S 2 ) such that p(Y ) ∈ V implies that p(X ) and p(Y ) are topologically equivalent; that is, there exists a homeomorphism of S 2 , which preserves S 1 , carrying orbits of the flow induced by p(X ) onto orbits of the flow induced by p(Y ), preserving sense but not necessarily parameterization.
Since in this paper we are interested in the classification of the structurally unstable quadratic vector fields of codimension two, we recall the concept of quadratic vector fields of lower codimension in structurally stability.
Recalling the works of Peixoto [22], restricted to the class of the quadratic vector fields, we have the following result: Theorem 1 Consider p(X ) ∈ P n (S 2 ) (or X ∈ P n (R 2 )). This system is structurally stable if and only if (i) the finite and infinite singular points are hyperbolic; (ii) the limit cycles are hyperbolic; (iii) there are no saddle connections. Moreover, the structurally stable systems form an open and dense subset of P n (S 2 ) (or P n (R 2 )).
The studies done up to now on structurally stable systems and codimension one systems are modulo limit cycles, so it is sufficient to consider only conditions (i) and (iii) of Theorem 1. We refer to these conditions as stable objects.
According to [3] there are 44 topologically distinct structurally stable quadratic vector fields. Concerning the codimension one quadratic vector fields, we allow the break of only one stable object. In other words, a quadratic vector field X is structurally unstable of codimension one modulo limit cycles if and only if (I) It has one and only one structurally unstable object of codimension one, i.e. one of the following types: (II) It has no structurally unstable limit cycles, saddle-point separatrices forming a loop, or singular points other than those listed in (I). (III) If the vector field has a saddle-node, none of its separatrices may go to a saddle point and no two separatrices of the saddle-node are continuation one of the other.
In what follows, instead of talking about codimension one modulo limit cycles, we will simply say codimension one * .
As described in Chapter 5 of [5], the codimension one * quadratic vector fields can be allocated in four sets, according to the bifurcations that occur to the singular points of structurally stable quadratic vector fields X .
(A) When a finite saddle and a finite node of X coalesce and disappear. (B) When an infinite saddle and an infinite node of X coalesce and disappear. (C) When a finite saddle (respectively node) and an infinite node (respectively saddle) of X coalesce and then they exchange positions. (D) When we have a saddle-to-saddle connection. This set is split into five subsets according to the type of the connection: (a) finite-finite (heteroclinic orbit), (b) loop (homoclinic orbit), (c) finite-infinite, (d) infinite-infinite between symmetric points and (e) infiniteinfinite between adjacent points.
Recalling the main result in [5], the phase portraits in all these four sets sum up 211 topological distinct ones, where 204 of these total are proved to be realizable and the remaining 7 are conjectured to be impossible.
However, in order to prove the main result of this paper, we went back to looking at the results in [5] and noticed an error in the realization of phase portrait U 1 A,49 . In fact, this phase portrait is not realizable, according to the next result.

Proposition 1 Phase portrait U 1
A,49 in [5] is not realizable (see this phase portrait in Fig. 2).

Remark 2
Despite the result presented by Proposition 1 eliminates phase portrait U 1 A,49 from the classification presented in [5], and that classification is the starting point of our study in this paper, we will keep phase portrait U 1 A,49 in our analysis and, in Sect. 4, we prove its impossibility and its consequence for the present study. Table 1 Sets of structurally unstable quadratic vector fields of codimension two considered from combinations of the sets of codimension one * : A, B, C and D (which in turn is split into a, b, c, d and e) Table 2 The next step is to classify, modulo limit cycles, the codimension two quadratic vector fields.
Since the concept of codimension applied to topological phase portraits of quadratic vector fields can become a little weird if we continue in this same way, we better give a better definition of codimension.

Definition 1
We say that a phase portrait of a quadratic vector field is structurally stable if any sufficiently small perturbation in the parameter space leaves the phase portrait topologically equivalent the previous one.

Definition 2
We say that a phase portrait of a quadratic vector field is structurally unstable of codimension k if any sufficiently small perturbation in the parameter space either leaves the phase portrait topologically equivalent the previous one or it moves it to a lower codimension one, and there is at least one perturbation that moves it to the codimension k − 1.
Remark 3 1. When applying these definitions, modulo limit cycles, to phase portraits with centers, it would say that some phase portraits with centers would be of codimension as low as two, while geometrically they occupy a much smaller region in R 12 . So, the best way to avoid inconsistencies in the definitions is to tear apart the phase portraits with centers, that we know they are in number 31 [32], and just work with systems without centers. 2. Starting in cubic systems, the definition of topologically equivalence, modulo limit cycles, becomes more complicated since we can have limit cycles having only one singularity in its interior or more than one. So we cannot collapse the limit cycle because its interior is also relevant for the phase portrait. 3. Moreover, our definition of codimension needs also more precision starting with cubic systems due to new phenomena that may happen there.
Then, according to this definition concerning codimension two, and the previously known results of codimension one, we have the result: is structurally unstable of codimension two modulo limit cycles if and only if all its objects are stable except for the break of exactly two stable objects. In other words, we allow the presence of two unstable objects of codimension one or one of codimension two.
Combining the sets of codimension one * quadratic vector fields one to each other, we obtain 10 new sets, where one of them is split into 15 subsets, according to Tables 1 and 2 .
Analogously, instead of talking about codimension two modulo limit cycles, we will simply say codimension two * .
Geometrically, the codimension two * sets can be described as follows. Let X be a codimension one * quadratic vector field. We have the following sets: Table 2 Sets of structurally unstable quadratic vector fields of codimension two in the set DD (see Table 1) Either when a finite saddle (respectively a finite node) of X coalesces with the finite saddle-node, giving birth to a semi-elemental triple saddle: s (3) (respectively a triple node: n (3) ), or when both separatrices of the saddle-node limiting its parabolic sector coalesce, giving birth to a cusp of multiplicity two: cp (2) , or when another finite saddlenode is formed, having then two finite saddle-nodes: sn (2) + sn (2) . Since the phase portraits with s (3) and with n (3) would be topologically equivalent to structurally stable phase portraits and we are mainly interested in new phase portraits, we will skip them in this classification. Anyway, we may find them in the papers [8] and [9]. (AB) When an infinite saddle and an infinite node of X coalesce plus a finite saddle-node: When we have a finite saddle-node and when a finite saddle (respectively node) and an infinite node (respectively saddle) of X coalesce: sn (2) + 1 1 S N . (AD) When we have a finite saddle-node plus a separatrix connection, considering all five types of set D. (BB) When an infinite saddle (respectively an infinite node) of X coalesces with an existing infinite saddle-node 0 2 S N , leading to a triple saddle: 0 3 S (respectively a triple node: 0 3 N ). This case is irrelevant to the production of new phase portraits since all the possible phase portraits that may be produced are topologically equivalent to a structurally stable one. (BC) When a finite antisaddle (respectively finite saddle) of X coalesces with an existing infinite saddle-node 0 2 S N , leading to a nilpotent elliptic saddle 1 2 E − H (respectively nilpotent saddle 1 2 H H H − H ). Or it may also happen that a finite saddle (respectively node) coalesces with an elemental node (respectively saddle) in a phase portrait having already an 0 2 S N , having then in total 1 1 S N + 0 2 S N . (BD) When we have an infinite saddle-node 0 2 S N plus a separatrix connection, considering all five types of set D.
(CC) This case has three possibilities: (i) when a finite saddle (respectively finite node) of X coalesces with an existing infinite saddle-node 1 1 S N , leading to a semi-elemental triple saddle 2 1 S (respectively a semi-elemental triple node 2 1 N ); (ii) when a finite saddle (respectively node) and an infinite node (respectively saddle) of X coalesce, plus another existing infinite saddle-node 1 1 S N , leading to two infinite saddle-nodes 1 1 S N + 1 1 S N ; (iii) when two finite saddles (respectively nodes) coalesce with an infinite node (respectively saddle) leading to a nilpotent saddle (respectively nilpotent elliptic saddle).
The first case is irrelevant to the production of new phase portraits since all the possible phase portraits that may be produced are topologically equivalent to a structurally stable one. (CD) When we have an infinite saddle-node 1 1 S N plus a saddle to saddle connection, considering all five types of set D.
(DD) When we have two saddle to saddle connections, which are grouped as follows: (aa) Two finite-finite heteroclinic connections; (ab) A finite-finite heteroclinic connection and a loop; (ac) A finite-finite heteroclinic connection and a finite-infinite connection; (ad) A finite-finite heteroclinic connection and an infinite-infinite connection between symmetric points; (ae) A finite-finite heteroclinic connection and an infinite-infinite connection between adjacent points; (bb) Two loops; (bc) A loop and a finite-infinite connection; (bd) A loop and an infinite-infinite connection between symmetric points; (be) A loop and an infinite-infinite connection between adjacent points; (cc) Two finite-infinite connections; (cd) A finite-infinite connection and an infinite-infinite connection between symmetric points; (ce) A finite-infinite connection and an infinite-infinite connection between adjacent points; (dd) Two infinite-infinite connections between symmetric points; (de) An infinite-infinite connection between symmetric points and an infinite-infinite connection between adjacent points; (ee) Two infinite-infinite connections between adjacent points. Some other of these cases have also been proved to be empty in an on course paper. The main goal of this paper is to present the global phase portraits of the vector fields X ∈ P 2 (R 2 ) belonging to the set A A and make sure that they are realizable. Let 2 0 denote the set of all planar structurally stable vector fields and 2 i (S) denote the set of all structurally unstable vector fields X ∈ P 2 (R 2 ) of codimension i, modulo limit cycles belonging to the set S, where S is a set of vector field with the same type of instability, for instance, X ∈ 2 2 (A A) denote the set of all structurally unstable vector fields X ∈ P 2 (R 2 ) of codimension two * belonging to the set A A.
With all of these we can formulate the next theorem.
, then its phase portrait on the Poincaré disc is topologically equivalent modulo orientation and modulo limit cycles to one of the 34 phase portraits of Fig. 1, and all of them are realizable.
In Sect. 2, we make a brief description of phase portraits of codimensions zero and one that are needed in this paper. In Sect. 3, we make the list of topologically possible phase portraits of codimension two in set A A, removing already some which are proved impossible, and in Sect. 4, we prove the realization of all of them but one, which is proved to be impossible with a more detailed argument. We also prove Proposition 1 in Sect. 4.

Quadratic Vector Fields of Codimension Zero and One
In this section we summarize all the needed results from the book of Artés et al. [5]. The following result is a restriction of Theorem 1.1 of [5] to the set A. We denote by 2 1 (A) the set of all structurally unstable vector fields X ∈ P 2 (R 2 ) of codimension one * belonging to the set A.
, then its phase portrait on the Poincaré disc is topologically equivalent modulo orientation and modulo limit cycles to one of the 70 phase portraits of Fig. 2, and all of them are realizable. The next result describes which phase portraits were discarded in [5] because they were not realizable, but their role now is important in the process of discarding impossible phase portraits of codimension two * .

Theorem 5
In order to obtain a phase portrait of a structurally unstable quadratic vector field of codimension one * from set A it is necessary and sufficient to coalesce a finite saddle and a finite node from a structurally stable quadratic vector field, which leads to a finite saddle-node, and after some small perturbation it disappears. For the vector fields in this set, the following statements hold.
(a) In Table 3 we may see in the first and fifth columns the structurally stable quadratic vector fields (following the notation present in [3,5]) which, after the coalescence of singularities cited above, lead to at least one phase portrait of codimension one * from set A. (b) Inside this set A, we have a total of 77 topologically distinct phase portraits according to the different α-limit or ω-limit of the separatrices of their saddles, 7 of which are non-realizable (they are given in Table 4). These numbers are given in the second and sixth columns of Table 3. (c) From these numbers of possible phase portraits, most of them are realizable. That is, even though there is the topological possibility of their existence, some of them break some analytical property which makes them not realizable inside quadratic vector fields.
We have a total of 70 realizable phase portraits. In the third and seventh columns of Table 3 we present the number of realizable cases coming from the bifurcation of each structurally stable phase portrait, and in the fourth and eighth columns we present the bifurcated phase portraits of codimension one * associated to each one. (d) There are then 7 non-realizable cases from set A which we now collect in a single picture (see Fig. 3) and denote by U 1 I ,b , where U 1 I stands for Impossible of codimension one * and b ∈ {1, 2, 3, 103, 104, 105, 106}. These phase portraits are all drawn in [5], distributed along the paper having already the notation given above. Anyway, we provide Table 4 in order to relate easily (giving also the page where they appear first and the page they are proved to be impossible).
An important result to study the impossibility of some phase portraits is Corollary 3.29 of [5].
Corollary 1 If one of the structurally stable vector fields that bifurcates from a possible structurally unstable vector field of codimension one is not realizable, then this unstable system is also not realizable.
This corollary can easily be adapted for higher codimensions. Theorem 6 If one of the phase portraits of codimension k that bifurcates from a possible codimension k + 1 phase portrait is not realizable, then this latter phase portrait is also not realizable.

Fig. 2 continued
Considering all the different ways to obtain phase portraits belonging to set A A of codimension two * , it is necessary to consider all possible ways of coalescing singular points. We split set A A into four subsets as follows: possessing a triple saddle s (3) , resulting from the coalescence of a finite saddle with the finite saddle-node in the direction of its center manifold; (A A n ) X ∈ 2 2 (A A) possessing a triple node n (3) , resulting from the coalescence of a finite node with the finite saddle-node in the direction of its center manifold; (A A cp ) X ∈ 2 2 (A A) possessing a cusp of multiplicity two cp (2) , resulting from the coalescence of the two separatrices of the saddle-node having the same stability; In this table, SSQVF stands for structurally stable quadratic vector fields, # p (respectively # r ) for the number of topologically possible (respectively realizable) phase portraits of codimension one * bifurcated from the respective SSQVF, and SU1 for the respective phase portraits of codimension one * The first and fourth columns indicate the structurally stable quadratic vector field (SSQVF) which suffers a bifurcation, the second and fifth columns indicate the pages where they appear in [5] and the third and sixth columns present the corresponding impossible phase portraits (A A snsn ) X ∈ 2 2 (A A) possessing two finite saddle-nodes sn (2) +sn (2) , resulting from the coalescence of a finite saddle with a finite node plus the existing finite saddle-node.
The next result is a useful tool when working with structurally unstable quadratic vector fields of codimension two * possessing a triple singular point (s (3) or n (3) ). Although it is stated for general polynomial vector fields, we will use it only for quadratic ones. (a) Any perturbation of X in a sufficiently small neighborhood of this point will produce either a structurally stable system (with two saddles and one node (respectively one saddle and two nodes), or with only one saddle (respectively one node) in the neighborhood), or a structurally unstable system of codimension one (with one saddle-node and one saddle (respectively one saddle-node and one node)), or a system topologically equivalent to X. (b) All these possibilities of structurally stable systems and of structurally unstable systems of codimension one * are realizable. (c) If the triple saddle s (3) (respectively triple node n (3) ) is the only unstable object of codimension two in the region of definition and we consider the perturbation which leaves a saddle-node and a saddle (respectively a saddle-node and a node) in the small neighborhood, then the parabolic sector of the saddle-node (respectively the node) is the ω-limit or α-limit (depending on its stability) of at least one of the separatrices of the saddle (respectively of the central manifold of the saddle-node). We will say that the saddle (respectively, the node) is linked with the saddle-node.
In order to complete the proof of this statement, we need to guarantee that this saddlenode can be either split into a saddle and a node or disappear, after applying a convenient perturbation. But this is done in Lemma 3.24 of [5]. Now, to prove statement (c), we recall that Lemma 3.24(c) of [5] assures that, after applying a convenient small perturbation to a saddle-node, it leaves a saddle and a node, in which case this node is the α-limit or ω-limit of at least one of the separatrices of the saddle. In this sense, having a triple saddle (respectively a triple node), from statement (a) above, there exists a perturbation which leaves two saddles and a node s 1 + n + s 2 (respectively a saddle and two nodes n 1 + s + n 2 ). Moreover, from this configuration of singular points, we can generate the following new configurations: s 1 n (2) +s 2 or s 2 n (2) +s 1 (respectively sn 1(2) +n 2 or sn 2(2) +n 1 ). Applying Lemma 3.24(c) of [5] to the saddle-node of each configuration, we obtain that the node n (respectively the saddle s) is linked to the saddles s 1 and s 2 (respectively the nodes n 1 and n 2 ). Then, we conclude that, after a perturbation of the triple saddle (respectively triple node), leading to a saddle-node and a saddle (respectively a saddle-node and a node), the parabolic sector of the saddle-node (respectively the node) is the α-limit or ω-limit of at least one of the separatrices of the saddle (respectively of the central manifold of the saddle-node).

Cases AA s and AA n
In the classes A A s and A A n , the unstable object of codimension two * is either a triple saddle s (3) or triple node n (3) .
By Lemma 1(c), the only way we can coalesce a saddle-node and a saddle or a node is by moving them towards one another along the orbit linking both of them. We will name provisionally the phase portraits which appear here as A A s b and A A n b , where b is a cardinal. Starting from a phase portrait of codimension one * of set A, we coalesce the saddle-node with the saddle (respectively the node), obtaining a phase portrait of codimension two * with a triple saddle (respectively, with a triple node), and then separating this point into a saddle (respectively a node) plus a saddle-node, we get a phase portrait of codimension one * also belonging to set A. Moreover, these unfoldings of codimension one * appear in pairs and each pair is linked by a single codimension two phase portrait. In the third column, we present the corresponding unfoldings of codimension one * Fig. 4, is topologically equivalent to one of the 44 structurally stable phase portraits in [3]. In Table 5 we present these equivalences, as well as the unfoldings of codimension one * .

Lemma 2 Each phase portrait from the classes A A s and A A n , shown in
Proof Using the technique of coalescing singular points, as in [3,5], we obtain all the topological phase portraits in Fig. 4.

Case AA cp
In the class A A cp , the unstable object of codimension two * is a cusp cp (2) . It is important to mention that here we are using the notation used in the book [7]. Starting from a phase portrait of codimension one * of set A, we coalesce the two separatrices of the saddle-node having the same stability, obtaining a phase portrait of codimension two * with a cusp, and then separating these separatrices, we get a phase portrait of codimension one * also belonging to set A. Moreover, these unfoldings of codimension one * appears in pairs in a one-to-one correspondence, that is, giving a phase portrait of codimension one * of set A, we can correspond one and only phase portrait of codimension one * by passing through the set A A cp . In order to do this coalescence of separatrices of the nodal sector of the saddle-node cannot receive any other separatrix. See for example phase portrait U 1 A,3 to find such impossibility.
All the phase portraits with a cusp were already studied in the paper of Jager [19], even of higher codimension than two and including other finite nilpotent singular points. So we could have relied on this paper and simply extract the codimension-two examples, but since we have found a gap in that paper and some phase portraits are missing (even though their are not of codimension two), we have preferred to obtain all the topological possibilities using a different proceeding and latter check that they fit with the results of Jager.
Phase portrait U 1 A,1 produces phase portrait A A cp 1 (see Fig. 5) and after bifurcation we get phase portrait U 1 A,1 . Phase portrait U 1 A,2 produces phase portrait A A cp 2 (see Fig. 6) and after bifurcation we get phase portrait U 1 A,9 . Phase portrait U 1 A,5 produces phase portrait A A cp 3 (see Fig. 7) and after bifurcation we get phase portrait U  . 19) and after bifurcation we get phase portrait U 1 A,69 . The remaining cases of codimension one * do not produce any phase portrait with a cusp since we cannot coalesce the separatrices of the saddle-node with the same stability without affecting other points, which produces a higher order codimension phase portrait. These 15 topologically different phase portraits with a cusp of codimension two * correspond exactly with the phase portraits of codimension two in [19]. See Table 8 (Sect. 4.2) which relates the phase portraits in AA cp # with the phase portraits of [19].

Case AA snsn
In the class A A snsn , the unstable object of codimension two * is the set of two finite saddlenodes sn (2) +sn (2) .
In order to obtain a phase portrait of codimension two * with two finite saddle-nodes starting from a phase portrait of codimension one * of set A, we keep the existing saddlenode p 1 and either build a new one p 2 by coalescing a saddle and a node, or add a new one.
On the other hand, from the phase portraits of codimension two * with two saddle-nodes, there exist two ways of obtaining phase portraits of codimension one * also belonging to set A after perturbation: making p 2 disappear or splitting each saddle-node p 1 and p 2 into a saddle and a node (see Remark 4). So it is not necessary to check the option of adding a saddle-node to a system already having one. We just need to seek systems A with sn (2) + s + a and coalesce the two elemental singularities.

Remark 4
We recall that, in quadratic differential systems, the finite singular points are zeroes of a polynomial of degree four. Since p 1 is already a singular point of multiplicity two, the remaining singular points are zeroes of a quadratic polynomial. In other words, they can be two simple singular points (a saddle and a node), a double point (saddle-node p 2 ) or two complex conjugate singular points (assuming that none of the finite singularities has gone to infinity).

Phase portrait U 1
A,2 produces phase portrait A A snsn 1 (see Fig. 20). After bifurcation we get phase portraits U 1 A,1 , by making the new saddle-node disappear, and U 1 A,4 , by splitting the original saddle-node into a saddle and a node.
Phase portrait U 1 A,3 cannot produce a coalescence with the elemental antisaddle and the elemental saddle because the elemental antisaddle is surrounded by the separatrices of the saddle-node, and so it cannot reach the saddle. This same situation will happen in other phase portraits, such as in U 1 A, 28 , and many others, and because it is quite simple to detect this phenomena, we will simply skip them.
The study of phase portrait U 1 A,4 is already contained in the study of U 1 A,2 . Phase portrait U 1 A,5 produces phase portrait A A snsn 2 (see Fig. 21). After bifurcation we get phase portraits U 1 A,1 , by making any saddle-nodes disappear, and U 1 A,5 , by splitting the original saddle-node into a saddle and a node. (see Fig. 22). After bifurcation we get phase portraits U 1 A,1 , by making any of the saddle-nodes disappear, and U 1 A,6 , by splitting the original saddle-node into a saddle and a node.
Phase portrait U 1 A,7 produces phase portrait A A snsn 4 (see Fig. 23). After bifurcation we get phase portraits U 1 A,1 , by making any of the saddle-nodes disappear, and U 1 A,7 , by splitting the original saddle-node into a saddle and a node.
Phase portrait U 1 A,8 produces phase portrait A A snsn 5 (see Fig. 24). After bifurcation we get phase portraits U 1 A,1 , by making any of the saddle-nodes disappear, and U 1 A,9 , by splitting the original saddle-node into a saddle and a node.
All the possibilities concerning U 1 A,9 are already contained in the study of U 1 A,8 . Phase portrait U 1 A, 22 produces phase portrait A A snsn 6 (see Fig. 25). After bifurcation we get phase portraits U 1 A, 21 , by making any of the saddle-nodes disappear, and U 1 A, 24 , by splitting the original saddle-node into a saddle and a node.
Phase portrait U 1 A,25 produces phase portrait A A snsn 7 (see Fig. 26). After bifurcation we get phase portraits U 1 A,21 , by making the new saddle-node disappear, U 1 A, 19 , by making the Even though phase portrait U 1 A,26 is going to produce an equivalent diagram as in Fig. 26, we will perform it to be sure of that, and we will avoid repeating this same case in the next similar steps. Phase portrait U 1 A,26 produces phase portrait A A snsn 7 (see Fig. 27). After bifurcation we get phase portraits U 1 A, 19 , by making the new saddle-node disappear, U 1 A,25 , by splitting the original saddle-node into a saddle and a node, and U 1 A,21 , by making the original saddle-node disappears.
Phase portrait U 1 A,27 produces the impossible phase portrait U 2 I ,1 (see Fig. 28), because by splitting the original saddle-node into a saddle and a node we obtain the impossible phase portrait U 1 I ,1 of codimension one * . Phase portrait U 1 A,29 produces phase portrait A A snsn 8 (see Fig. 29). After bifurcation we get phase portraits U 1 A,21 , by making the new saddle-node disappear, U 1 A,30 , by splitting the original saddle-node into a saddle and a node, and U 1 A,20 , by making the original saddle-node disappear.
Phase portrait U 1 A,31 produces phase portrait A A snsn 9 (see Fig. 30). After bifurcation we get phase portraits U 1 A, 19 , by making any of the saddle-nodes disappear, and U 1 A,33 , by splitting the original saddle-node into a saddle and a node. Phase portrait U 1 A,34 produces phase portrait A A snsn 10 (see Fig. 31). After bifurcation we get phase portraits U 1 A, 19 , by making the new saddle-node disappear, U 1 A,35 , by splitting the original saddle-node into a saddle and a node, and U 1 A,20 , by making the original saddle-node disappear.
Phase portrait U 1 A,36 produces phase portrait A A snsn 11 (see Fig. 32). After bifurcation we get phase portraits U 1 A,20 , by making the new saddle-node disappear, U 1 A,38 , by splitting the original saddle-node into a saddle and a node, and U 1 A,21 , by making the original saddle-node disappear Phase portrait U 1 A,39 produces phase portrait A A snsn 12 (see Fig. 33). After bifurcation we get phase portraits U 1 A, 21 , by making any of the saddle-nodes disappear, and U 1 A,39 , by splitting the original saddle-node into a saddle and a node.
Phase portrait U 1 A,40 produces phase portrait A A snsn 13 (see Fig. 34). After bifurcation we get phase portraits U 1 A, 19 , by making the new saddle-node disappear, U 1 A,41 , by splitting the original saddle-node into a saddle and a node, and U 1 A,20 , by making the original saddle-node disappear.
Phase portrait U 1 A,42 produces the impossible phase portrait U 2 I ,2 (see Fig. 35), because by splitting the original saddle-node into a saddle and a node we obtain the impossible phase portrait U 1 I ,2 of codimension one * . Phase portrait U 1 A,44 produces phase portrait A A snsn 14 (see Fig. 36). After bifurcation we get phase portraits U 1 A,20 , by making any of the saddle-nodes disappear, and U 1 A,44 , by splitting the original saddle-node into a saddle and a node.
Phase portrait U 1 A,45 produces phase portrait A A snsn 15 (see Fig. 37). After bifurcation we get phase portraits U 1 A,21 , by making the new saddle-node disappear, U 1 A,46 , by splitting the original saddle-node into a saddle and a node, and U 1 A, 19 , by making the new saddle-node disappear.
Phase portrait U 1 A,47 produces phase portrait A A snsn 16 (see Fig. 38). After bifurcation we get phase portraits U 1 A,21 , by making the new saddle-node disappear, U 1 A,50 , by splitting the original saddle-node into a saddle and a node, and U 1 A,20 , by making the original saddle-node disappear.
Phase portrait U 1 A,48 produces phase portrait A A snsn 17 (see Fig. 39). After bifurcation we get phase portraits U 1 A,21 , by making the new saddle-node disappear, U 1 A,49 , by splitting the original saddle-node into a saddle and a node, and U 1 A,20 , by making the original saddle-node disappear.
Phase portrait U 1 A,51 produces phase portrait A A snsn 18 (see Fig. 40). After bifurcation we get phase portraits U 1 A, 19 , by making the new saddle-node disappear, U 1 A,53 , by splitting the original saddle-node into a saddle and a node, and U 1 A,21 , by making the new saddle-node disappear.
Phase portrait U 1 A,52 produces the impossible phase portrait U 2 I ,3 (see Fig. 41), because by splitting the original saddle-node into a saddle and a node we obtain the impossible phase portrait U 1 I ,3 of codimension one * . Phase portrait U 1 A,54 produces phase portrait A A snsn 19 (see Fig. 42). After bifurcation we get phase portraits U 1 A, 19 , by making any of the saddle-nodes disappear, and U 1 A,54 , by splitting the original saddle-node into a saddle and a node.
Phase portrait U 1 A,55 produces phase portrait A A snsn 20 (see Fig. 43). After bifurcation we get phase portraits U 1 A, 19 , by making any of the saddle-nodes disappear, and U 1 A,55 , by splitting the original saddle-node into a saddle and a node. The remaining cases of codimension one * do not produce any phase portrait with two saddle-nodes since either (1) they do not have enough finite singular points to produce another saddle-node, or (2) the saddle and the node are not directly linked. See Table 6 for the corresponding cases.  Table 6 Codimension one * phase portraits that do not produce any phase portrait with two saddle-nodes according to their respective reason In the first column we present the reasons and in the second one we list the corresponding cases

Introduction
In the previous section we have produced all the topologically possible phase portraits for structurally unstable quadratic systems of codimension two * belonging to the set 2 2 (A A). And from them, we have already discarded some which are not realizable due to their unfoldings of codimensions one and zero are impossible. The data is summarized in Table 7.
In this section we prove that one case from A A snsn is not realizable and we give specific examples for the 46 different topological classes of structurally unstable quadratic systems of codimension two * .
In [3] the authors point out that all 44 structurally stable phase portraits could be obtained without limit cycle and they prove this one by one. On the contrary, due to the large number of cases, in [5] the authors did not follow the same procedure for the 204 structurally unstable phase portraits of codimension one * . Since the present paper is directly derived from this second study, we have found examples with no signals of limit cycles, but we have not proved the absence of infinitesimal ones.
In the attempt of seeking for concrete examples of each of the unstable systems of codimension two * previously found, we have relied on many papers where families of quadratic systems had been studied, so that either from themselves, or by a perturbation of them, the wanted phase portraits appeared. More concretely, the useful papers have been: (1) [8] where the set of all real quadratic polynomial differential systems with a finite semielemental triple saddle was topologically classified, and by using the phase portraits of generic regions we realize the cases of set A A s . (2) [9] where the set of all real quadratic polynomial differential systems with a finite semielemental triple node was topologically classified, and by using the phase portraits of generic regions we realize the cases of set A A n . (3) [19] where the author classified all quadratic systems with a cusp, and by using directly some phase portraits of Jager's classification we realize the cases of set A A cp . (4) [10] where the set of all real quadratic polynomial differential systems with a finite saddlenode and an infinite saddle-node 0 2 S N were topologically classified, and by using the theory rotated vector fields on systems from surface S 2 (where another finite saddle-node exists) we may either break the infinite saddle-node into elemental singular points, or making it disappear, we produce the cases of set A A snsn .
Using these papers we could find all possible examples from the four sets we study here. For the cases A A s and A A n , because of Lemma 2, we do not show the realization of such phase portraits here. In the next two sections we show the realization of phase portraits of cases A A cp and A A snsn .

Realization of Cases AA cp
Now we give examples of all realizable structurally unstable phase portraits of codimension two * for quadratic systems having a cusp. Although there exist different papers having exam- Table 8 Correspondence between codimension two * phase portraits of set A A c and the phase portraits in [19]. In the first column we present the definitive notation of the realizable phase portraits, in the second column we present the codimension two * phase portraits of set A A c in the present paper, in the third column we show the corresponding phase portraits in [19], in the fourth column we specify the corresponding normal form and in the other columns we present the values of the parameters of (3) and (4) which realizes such phase portrait Cod 2 * Ref. [19] Normal form Fig. 12 (3) 12c, Fig. 22 (4) 1 4 −10 4/5 ples realizing these phase portraits, we chose the paper [19] from which we can obtain all of them directly.
In [19] there are many phase portraits which produce a phase portrait of set A A cp . In Table 8 we simply present one representative from generic regions of the bifurcation diagram of (3) and (4) corresponding to the phase portrait of codimension two * .

Realization of Cases AA snsn
In this section we provide examples of the realizable structurally unstable phase portraits of codimension two * for quadratic systems having two finite saddle-nodes. In opposite to the previous cases, as far as we know, this type of family of quadratic systems has not been topologically classified, so that we do not count with a paper which provides the desired phase portraits of codimension two * in a direct way.
In [10] the authors studied the geometry of the quadratic systems possessing a finite saddle-node sn (2) and an infinite saddle-node 0 2 S N . In the bifurcation diagram described in [10], the surface of bifurcation S 2 consists on the systems with two finite saddle-nodes and the infinite saddle-node 0 2 S N . Moreover, we observe that if we apply some perturbation on systems belonging to this surface S 2 that splits the infinite saddle-node into a saddle and a node (both infinite) and keeps untouched both finite saddle-nodes, we obtain all but one of the realizable cases of set A A snsn . Thus, the way of providing these examples is considering a rotated family of vector fields.
First, we prove that 19 cases are realizable using perturbations of phase portraits from [10]. We consider systeṁ with g, h, l and n real constants, which is the normal form in [10] of quadratic systems possessing a finite saddle-node sn (2) and an infinite saddle-node 0 2 S N located at the bisector of the first and third quadrants.
In Table 9 we present the coefficients of system (6) which has the phase portraits of set A A snsn , derived from the rotation of systems (5) on surface S 2 from their bifurcation diagram. Now, we proceed to prove the impossibility of phase portrait A A snsn 17 . Before we get this conclusion, we prove Proposition 1.

Proof of Proposition 1: Correcting a Mistake in [5]
In this section we point out a mistake that the authors in [5] made and its correction implies in the non-realization of phase portrait A A snsn 17 . In [5] it is proved that structurally unstable quadratic differential systems of codimension one modulo limit cycles have up to 204 topologically different phase portraits (and at most 211, but these extra seven phase portrait are conjectured to be non-realizable).
However, there exists a mistake in the book, since the example given for U 1 A,49 is wrong and, moreover, this phase portrait is not realizable as we will prove here.
The new proof is based mainly on Lemmas 3.1 and 3.17 and Corollary 3.7 of [5] plus the next two lemmas. In the first column we present the definitive notation of the realizable phase portraits, in the second column we present the codimension two * phase portraits of set A A snsn in the present paper, in the third column we show the derived phase portrait in [10] before rotation and in the other columns, the coefficients of system (6) Lemma 3 A finite saddle of a quadratic system whose separatrices are not invariant straight lines, have both stable (respectively unstable) separatrices having the same stability on the same semi-plane delimited by the straight line formed by the eigenvector of the Jacobian matrix associated to the negative (respectively positive) eigenvalue.
Proof Assuming the contrary, then we would have a local phase portrait around the saddle as in Fig. 44, and a straight line close to the line tangent to the separatrices would have three contact points (including the singular point).

Lemma 4
An infinite saddle of a quadratic system whose finite separatrices do not form an invariant straight line have both of its finite separatrices tangent to a straight line defined by the eigenvector (in the affine chart). Moreover, even these saddles at infinity are in fact saddles of a cubic system, they also have both separatrices on the same side of this straight line on the infinite local chart. Once translated to the affine local chart, they stay on different semi-planes. Proof The fact that the separatrices of infinite saddles remain tangent to a single straight line in the affine chart comes simply from the Poincaré compactification (see Fig. 45a).
If the separatrices in the infinite chart stay on the same semi-plane as defined by the eigenvector (as it happens with separatrices of finite saddles of quadratic systems), then the finite separatrices of infinite saddles stay in the affine plane in different semi-planes regarding the straight line defined by the eigenvector (see again Fig. 45a, b). But being separatrices of a cubic system, they do not need to hold Lemma 3. Anyway, since they are separatrices of a cubic system which is the extension of a quadratic one under the Poincaré compactification, they must continue respecting Lemma 3, otherwise, on the affine chart they would be placed on the same side of the semi-plane, and then their curvature would produce two contact points with a parallel line close to the one defined by the eigenvector (see Fig. 45c).
Since we will strongly need Lemma 3.17 from [5], we will repeat it here and also the picture that gives its proof.

Lemma 5
The αand ω-limit of the four separatrices of a finite elementary saddle in a quadratic system cannot stay in the same open semi-plane delimited by a straight line passing through the saddle. (See Fig. 46.) Now, in order to prove the impossibility of U 1 A,49 , we must start from the structurally stable phase portrait from which it comes from, namely S 2 10,13 . Phase portrait S 2 10,13 has two finite saddles and two finite antisaddles, plus 3 infinite singularities (see Fig. 47). Phase portrait S 2 10,13 is a very tricky one. In fact, it was the last one which was found among the structurally stable ones since it does not come from a simple perturbation of a system with a cusp (or with an invariant straight line) as many other structurally stable phase portraits, but it is necessary two different perturbations to obtain it. Moreover, it seems to live in a very small region of the parameter space. Anyway, its existence is beyond doubt and the example given in [3] is good. We reproduce it here: x = y + 3x 2 + 11x y/8, y = ε 2 x/5 − εy + x 2 + 3x y + 5y 2 /4 + α(y + 3x 2 + 11x y/8), with ε = 0.2 and α = −0.017.
Moreover, the way we draw phase portrait S 2 10,13 in Fig. 47 seems to contradict Lemma 3.17 from [5], and in fact it does. The real numerical image we obtain is topologically equivalent to this, but difficult to grasp.
So, we will give now an argument so that S 2 10,13 can be drawn respecting Lemma 5 (Lemma 3.17 from [5]).
In Fig. 48, we start by drawing the phase portrait having the infinite saddle at [1:0:0] and one finite saddle at the origin. This can always be done by means of a translation and a rotation. We draw in dashed lines the horizontal axis and the straight line defined by the eigenvector of the infinite saddle. We call L to this line. They cannot coincide since both separatrices from infinite have to split the plane in three regions leaving all finite singularities in the same region which cannot be the central one (see Fig. 48a) and this is not possible if both separatrices stay on different semi-planes with respect to the horizontal axis. Moreover, the separatrices of the finite saddle cannot be tangent to the horizontal axis, because otherwise a parallel line to the axis, close to it, would produce too many contact points.
So, we have the starting position given by Fig. 48b. We can assume one direction for one of the infinite separatrices, and then the direction of the other infinite separatrix and the line at infinity in Fig. 48b is forced. We cannot guess the stabilities of the separatrices of the finite saddle yet.
The flow on the horizontal line L must always point down, since it goes down on the right and on the left and it already has a contact point on the infinite singularity.  The affine plane is divided in three horizontal bands and inside the lower one we will have only the separatrix from the left infinite saddle which must go to an infinite unstable node. Between this saddle and the infinite unstable node, there must exist an infinite stable node. In order to obtain phase portrait S 2 10,13 the stable infinite separatrix must come from the unstable infinite node on the upper half disk. So we arrive at Fig. 49. We have drawn this separatrix close to the line L just to give space for the rest of the picture. Moreover, this separatrix cannot cross L in order to avoid more contact points.
Phase portrait S 2 10,13 shows an interesting combination of separatrices. We can see as each finite saddle sends (or receives) a separatrix from an infinite singular point, then sends the opposite separatrix to a finite antisaddle which receives another separatrix from the other finite saddle, and the opposite separatrix to this last one goes to the original infinite singular point. And this happens not just once but twice. In order to give a name to this chain of separatrices, we will call it a tear, and the two tears linked in this way will be named chained tears.
So we have one tear which will come from the upper stable node and a second tear from the upper unstable one. On the crossing point of these two tears (apart from the origin) we will have the second finite saddle. For the moment we do not worry about where the antisaddles are. We have two ways to draw this: either the infinite stable node will use the separatrices α's or the β's. So we have the two possibilities drawn in Fig. 50.
In the first case, the second finite saddle stays in the lower half disk; in the second, in the upper half disk. The antisaddles must be then located in the middle of the curves that join both saddles splitting them in two separatrices each, and the orientation of all the separatrices is forced.  Even case (B) seems more natural and less struggled, it is in fact impossible to be realized since it clearly violates Lemma 5. So, case (A) is the only possibility. But even case (A) also seems to violate Lemma 5, not with respect to the saddle at the origin but to the other saddle. Since we know that phase portrait S 2 10,13 is realizable, and this is the only possible scheme, we must be able to provide an explanation for this.
Let us move this second saddle to the origin by means of a translation. So the saddle now at the origin will move to the upper semi-disk. Then we obtain Fig. 51.
The stable finite antisaddle must be in the upper semi-disk, but we have not yet drawn the finite unstable antisaddle on purpose. If we draw it also in the upper semi-disk, then the whole phase portrait will violate Lemma 5. The only way to avoid this violation is to place the unstable antisaddle on the lower semi-disk as in Fig. 52. The only difference between Figs. 51 and 52 is that we have added the last finite antisaddle.