First integrals and phase portraits of planar polynomial differential cubic systems with invariant straight lines of total multiplicity eight

In [C. Bujac, J. Llibre, N. Vulpe, Qual. Theory Dyn. Syst. 15(2016), 327–348] all first integrals and phase portraits were constructed for the family of cubic differential systems with the maximum number of invariant straight lines, i.e. 9 (considered with their multiplicities). Here we continue this investigation for systems with invariant straight lines of total multiplicity eight. For such systems the classification according to the configurations of invariant lines in terms of affine invariant polynomials was done in [C. Bujac, Bul. Acad. S, tiint,e Repub. Mold. Mat. 75(2014), 102–105], [C. Bujac, N. Vulpe, J. Math. Anal. Appl. 423(2015), 1025–1080], [C. Bujac, N. Vulpe, Qual. Theory Dyn. Syst. 14(2015), 109–137], [C. Bujac, N. Vulpe, Electron. J. Qual. Theory Differ. Equ. 2015, No. 74, 1–38], [C. Bujac, N. Vulpe, Qual. Theory Dyn. Syst. 16(2017), 1–30] and all possible 51 configurations were constructed. In this article we prove that all systems in this class are integrable. For each one of the 51 such classes we compute the corresponding first integral and we draw the corresponding phase portrait.


Introduction
Polynomial differential systems on the plane are systems of the form ẋ = P(x, y), ẏ = Q(x, y), (1.1) where P, Q ∈ R[x, y], i.e.P and Q are the polynomials over R. To a system (1.1) we can associate the vector field We call cubic a differential system (1.1) with degree n = max{deg P, deg Q} = 3.
There are several open problems on polynomial differential systems, especially on the class of all cubic systems (1.1) (denote by CS the whole class of such systems).In this paper we are concerned with questions regarding integrability in the sense of Darboux and classification of all phase portraits of CS.These problems are very hard even in the simplest case of quadratic differential systems.
The method of integration of Darboux uses multiple-valued complex functions of the form: and f i irreducible over C. It is clear that in general the last expression makes sense only for G 2 = 0 and for points (x, y) Consider the polynomial system of differential equations (1.1).The equation f (x, y) = 0 ( f ∈ C[x, y], where C[x, y] denotes the ring of polynomials in two variables x and y with complex coefficients) which describes implicitly some trajectories of systems (1.1), can be seen as an affine representation of an algebraic curve of degree m.Suppose that (1.1) has a solution curve which is not a singular point, contained in an algebraic curve f (x, y) = 0.It is clear that the derivative of f x(t), y(t) with respect to t must vanish on the algebraic curve f (x, y) = 0, so In 1878 Darboux introduced the notion of the invariant algebraic curve for differential equations on the complex projective plane.This notion can be adapted for systems (1.1).According to [13] the next definition follows.
Definition 1.1.An algebraic curve f (x, y) = 0 in C 2 with f ∈ C[x, y] is an invariant algebraic curve (an algebraic particular integral) of a polynomial system (1.1) if X( f ) = f K for some polynomial K(x, y) ∈ C[x, y] called the cofactor of the invariant algebraic curve f (x, y) = 0.
In view of Darboux's definition, an algebraic solution of a system of equations (1.1) is an invariant algebraic curve f (x, y) = 0, f ∈ C[x, y] (deg f ≥ 1) with f an irreducible polynomial over C. Darboux showed that if a system (1.1) possesses a sufficient number of such invariant algebraic solutions f i (x, y) = 0, f i ∈ C, i = 1, 2, . . ., s, then the system has a first integral of the form (1.3).
In 1979 Jouanolou proved the next theorem which completes part II of Darboux's Theorem.
The following theorem from [19] improves the Darboux theory of integrability and the above result of Jouanolou taking into account not only the invariant algebraic curves (in particular invariant straight lines) but also their algebraic multiplicities.We mention here this result adapted for two-dimensional vector fields.
Theorem 1.4 ([12, 19]).Assume that the polynomial vector field X in C 2 of degree d > 0 has irreducible invariant algebraic curves.
(i) If some of these irreducible invariant algebraic curves have no defined algebraic multiplicity, then the vector field X has a rational first integral.
(ii) Suppose that all the irreducible invariant algebraic curves f i = 0 have defined algebraic multiplicity q i for i = 1, . . ., p.If X restricted to each curve f i = 0 having multiplicity larger than 1 has no rational first integral, then the following statements hold.
(a) If ∑ p i=1 q i ≥ N + 1, then the vector field X has a Darboux first integral, where N = ( 2+d−1 2 then the vector field X has a rational first integral. We note that the notion of "algebraic multiplicity" of an algebraic invariant curve is given in [12] where in particular the authors proved the equivalence of "geometric" and "algebraic" multiplicities of an invariant curve for the polynomial systems (1.1).
If f (x, y) = ux + vy + w = 0, (u, v) = (0, 0) and X( f ) = f K where K(x, y) ∈ C[x, y], then f (x, y) = 0 is an invariant line of the family of systems (1.1).We point out that if we have an invariant line f (x, y) = 0 over C it could happen that multiplying the equation by a number λ ∈ C * = C \ {0}, the coefficients of the new equation become real, i.e. (uλ, vλ, wλ) ∈ R 3 ).In this case, along with the curve f (x, y) = 0 (sitting in in C 2 ) we also have an associated real curve (sitting in R 2 ) defined by λ f (x, y).
Note that, since a system (1.1) is real, if its associated complex system has a complex invariant straight line ux + vy + w = 0, then it also has its conjugate complex invariant straight line ūx + vy + w = 0.
To a line f (x, y) = ux + vy + w = 0, (u, v) = (0, 0) we associate its projective completion The line Z = 0 in P 2 (C) is called the line at infinity of the affine plane C 2 .It follows from the work of Darboux (see, for instance, [13]) that each system of differential equations of the form (1.1) over C yields a differential equation on the complex projective plane P 2 (C) which is the compactification of the differential equation Qdx − Pdy = 0 in C 2 .The line Z = 0 is an invariant manifold of this complex differential equation.
For an invariant line f (x, y) = ux + vy + w = 0 we denote â = (u, v, w) ∈ C 3 .We note that the equation λ f (x, y) = 0 where λ ∈ C * and C * = C\{0} yields the same locus of complex points in the plane as the locus induced by f (x, y) = 0.So that a straight line defined by â can be identified with a point [ â] = [u : v : w] in P 2 (C).We say that a sequence of straight lines f i (x, y) = 0 converges to a straight line f (x, y) = 0 if and only if the sequence of points [a i ] converges to [ â] = [u : v : w] in the topology of P 2 (C).Definition 1.5 ([27]).We say that an invariant affine straight line f (x, y) = ux + vy + w = 0 (respectively the line at infinity Z = 0) for a real cubic vector field X has multiplicity m if there exists a sequence of real cubic vector fields X k converging to X, such that each X k has m (respectively m − 1) distinct (complex) invariant affine straight lines 3 , converging to f = 0 as k → ∞, in the topology of P 2 (C), and this does not occur for m + 1 (respectively m).
We first remark that in the above definition we made an abuse of notation.Indeed, to talk about a complex invariant curve we need to have a complex system.However we said that the real systems X k meaning of course the complex systems associtated to the real ones X k .
We remark that the above definition is a particular case of the definition of geometric multiplicity given in paper [12], and namely the notion of "strong geometric multiplicity" with the restriction, that the corresponding perturbations are cubic systems.
The existence of sufficiently many invariant straight lines of planar polynomial systems could be used for integrability of such systems.During the past 15 years several articles were published on this theme.Investigations concerning polynomial differential systems possessing invariant straight lines were done by Popa, Sibirski, Llibre, Gasull, Kooij, Sokulski, Zhang Xi Kang, Schlomiuk, Vulpe, Dai Guo Ren, Artes as well as Dolov and Kruglov.
According to [1] the maximum number of invariant straight lines taking into account their multiplicities for a polynomial differential system of degree m is 3m when we also consider the straight line at infinity.This bound is always reached if we consider the real and the complex invariant straight lines, see [12].
So the maximum number of the invariant straight lines (including the line at infinity Z = 0) for cubic systems with finite number of infinite singularities is 9.A classification of all cubic systems possessing the maximum number of invariant straight lines taking into account their multiplicities has been made in [18].The authors used the notion of configuration of invariant lines for cubic systems (as introduced in [27], but without indicating the multiplicities of real singularities) and detected 23 such configurations.Moreover in this paper using invariant polynomials with respect to the action of the group Aff (2, R) of affine transformations and time rescaling (i.e.Aff (2, R) × R * )), the necessary and sufficient conditions for the realization of each one of 23 configurations were detected.A new class of cubic systems omitted in [18] was constructed in [4].Definition 1.6 ([31]).Consider a real planar cubic system (1.1).We call configuration of invariant straight lines of this system, the set of (complex) invariant straight lines (which may have real coefficients) of the system, each endowed with its own multiplicity and together with all the real singular points of this system located on these invariant straight lines, each one endowed with its own multiplicity.
The configurations of invariant straight lines which were detected for various families of systems (1.1) using Poincaré compactification, could serve as a base to complete the whole Poincaré disc with the trajectories of the solutions of corresponding systems, i.e. to give a full topological classification of such systems.For example, in papers [28,30] for quadratic systems with invariant lines greater than or equal to 4, it was proved that we have a total of 57 distinct configurations of invariant lines which leads to the existence of 135 topologically distinct phase portraits.In [25,26,35,36] the existence of 113 topologically distinct phase portraits was proved for cubic systems with invariant lines of total parallel multiplicity six or seven, taking in consideration the configurations of invariant lines of these systems.The notion of "parallel multiplicity" could be found in [36].
In this paper we consider the analogous problems for a specific class of cubic systems which we denote by CSL 8 .We say that a cubic system belongs to the family CSL 8 if it possesses invariant straight lines of total multiplicity 8, including the line at infinity and considering their multiplicities.
The goal of this article is to complete the study we began in [5][6][7][8][9].More precisely in this work we • prove that all systems in the class CSL 8 are integrable.We show this by using the geometric method of integration of Darboux.We construct explicit Darboux integrating factors and we give the list of first integrals for each system in this class; • construct all possible phase portraits of the systems in this class and prove that only 30 of them are topologically distinct; • give invariant (under the action of the group Aff (2, R) × R * )) necessary and sufficient conditions, in terms of the twenty coefficients of the systems, for the realization of each specific phase portrait.
This article is organized as follows.
In Section 2 we give the list of affine invariant polynomials and some notion and results needed in this article.
In Section 3 we present some preliminary results.More exactly, in Theorem 3.1 we describe all the 51 possible configurations of invariant lines which could possess the cubic systems in the class CSL 8 .Moreover we give necessary and sufficient conditions for the realization of each of these configurations.These results (obtained in [5][6][7][8][9]) serve as a base for the construction of the phase portraits as well as for determining of the corresponding first integrals and integrating factors.
Section 4 contains the main results of this article formulated in the Main Theorem.In Table 4.1 we give the canonical forms of systems in CSL 8 as well as the corresponding first integrals and integrating factors.We prove that each one of the 51 configurations given by Theorem 3.1 leads to a single phase portrait, except the configuration Config.8.6, which generates two topologically distinct phase portraits.In Table 4.1 we also present the necessary and sufficient affine invariant conditions for the realization of each one of the phase portraits obtained.Defining some geometric invariants, we prove (see Diagram 4.1) that among the obtained 52 phase portraits only 30 of them are topologically distinct.

Invariant polynomials associated with cubic systems possessing invariant lines
As it was mentioned earlier our work here is based on the result of the papers [4,[6][7][8][9] where the classification theorems according to the configurations of invariant straight lines for different subfamilies (i.e.systems with either 4 or 3 or 2 or 1 infinite distinct singularities) of systems in CSL 8 were proved (see further below).In what follows we recall some results in [18] which will be needed to state the mentioned theorems.Consider real cubic systems, i.e. systems of the form: ẋ = p 0 + p 1 (x, y) + p 2 (x, y) + p 3 (x, y) ≡ P(x, y), with real coefficients and variables x and y.The polynomials p i and q i (i = 0, 1, 2, 3) are the following homogeneous polynomials in x and y: It is known that on the set CS of all cubic differential systems (2.1) acts the group Aff (2, R) of affine transformations on the plane [27].For every subgroup G ⊆ Aff (2, R) we have an induced action of G on CS.We can identify the set CS of systems (2.1) with a subset of R 20  For the definitions of an affine or GL-comitant or invariant as well as for the definition of a T-comitant and CT-comitant we refer the reader to [27].Here we shall only construct the necessary Tand CT-comitants associated to configurations of invariant lines for the family of cubic systems mentioned in the statement of Main Theorem.
Let us consider the polynomials As it was shown in [33] the polynomials of degree one in the coefficients of systems (2.1) are GL-comitants of these systems.[16,22]) Here f (x, y) and g(x, y) are polynomials in x and y of the degrees r and s, respectively, and a ∈ R 20 is the 20-tuple formed by all the coefficients of system (2.1).
We remark that the set of GL-invariant polynomials (2.2) could serve as bricks for the construction of any GL-invariant polynomial of an arbitrary degree.More precisely as it was proved in [37] we have the next result.
In order to define the needed invariant polynomials it is necessary to construct the following GL-comitants of second degree with respect to the coefficients of the initial systems: , .
Next we consider the differential operator L = x • L 2 − y • L 1 constructed in [3] and acting on R[a, x, y], where .
(i) The total multiplicity of all finite singularities of this system equals 9 − k if and only if for every i ∈ {0, 1, . . ., k − 1} we have µ i ( ã, x, y) = 0 in the ring R[x, y] and µ k ( ã, x, y) = 0.In this case the factorization µ k ( ã, x, y) = ∏ k i=1 (u i x − v i y) = 0 over C indicates the coordinates [v i : u i : 0] of those finite singularities of the system (S) which "have gone" to infinity.Moreover the number of distinct factors in this factorization is less than or equal to four (the maximum number of infinite singularities of a cubic system) and the multiplicity of each one of the factors u i x − v i y gives us the number of the finite singularities of the system (S) which have coalesced with the infinite singular point [v i : u i : 0].

Preliminary results: the classification theorem for the family of systems in CSL 8
As it was mentioned in Section 2 our work is based on the results of the papers [4,[6][7][8][9] where the classification theorems according to the configurations of invariant straight lines for different subfamilies (i.e.systems with either 4 or 3 or 2 or 1 infinite distinct singularities) of systems in CSL 8 were proved.More precisely, our results could be described as follows: • In [6] the investigation of the subfamily of cubic systems in CSL 8 possessing 4 distinct infinite singularities was done.As a result it was proved that a system in this class could possess only one of the 17 configurations Config.8.1-Config.8.17 given in Figure 3.1.
• The subfamily of cubic systems in CSL 8 possessing 3 distinct infinite singularities was considered in [7].It was proved that a system in this class could possess only one of the 5 configurations Config.8.18-Config.8.22 presented in Figure 3.1.
• In the articles [5] and [8] the subfamily of cubic systems in CSL 8 with 2 distinct infinite singularities was investigated.This class contains cubic systems which could possess one of the 25 configurations Config.8.23-Config.8.47 given in Figure 3.1.
• And finally in [9] were examined the cubic systems in CSL 8 possessing a single infinite singular point (which is real).It was detected exactly 8 configurations Config.8.48-Config.8.51 (see Figure 3.1) which could possess a cubic system belonging to this class.
We join here all these results (formulated as Main Theorems in the above mentioned articles) in the following classification theorem.Theorem 3.1.Assume that a cubic system (1.1) is non-degenerate, i.e. ∑ 9 i=0 µ 2 i = 0. Then this system belongs to the family CSL 8 , i.e. it possesses invariant straight lines of total multiplicity 8 (including the line at infinity with its own multiplicity), if and only if one of the sets of the conditions Cond.1-Cond.51 given in Table 3.1 is satisfied.In addition, this system possesses exactly one of the 51 configurations Config.8.j (j ∈ {1, . . .51}) of invariant straight lines shown in Figure 3.1.Furthermore the quotient set under the action of the affine group and time rescaling on CSL 8 is formed by: (i) a discrete set of 22 orbits; (ii) a set of 29 one-parameter families of orbits.A system of representatives of the quotient set is given in Table 4.1 (column 1).

Main results
In this section we state and prove the main results of this article.
Main Theorem.Consider a non-degenerate cubic system (2.1), i.e. the condition ∑ 9 i=0 µ 2 i = 0 holds and assume that it belongs to the family CSL 8 .More precisely we assume that this system possesses one of the configurations Config.8.j (j = 1, . . ., 51) (see Figure 3.1), i.e. the corresponding set of the conditions Cond.j given in Table 3.1 is satisfied.Then: (A) this system is integrable and it has the first integral F j of generalized Darboux type (1.3)and the corresponding rational integrating factor R j ∈ R(x, y) (j = 1, . . ., 51) as it is indicated in Table 4.1 (column 3).This table also lists the corresponding invariant lines and their multiplicities, see column 2; (B) the phase portrait of this system corresponds to one of the 52 phase portraits P. 8.1-P.8.5, P. 8.6(a), P. 8.6(b), P. 8.7-P.8.51 (see Figure 4.1) if and only if the associated affine invariant conditions given in Table 4.1 (column 4) are satisfied.
(C) Among the 52 phase portraits given in Figure 4.1 there are exactly 30 topologically distinct phase portraits as at is indicated in Diagram 4.1 using the geometric invariants defined in Remark 4.2.
Corollary 4.1.All the systems in CSL 8 have elementary real first integrals.We only list below in Table 4.2 all real first integrals which correspond to those in the column 3 of Table 4.1 which are given there in complex form.Remark 4.2.In order to distinguish topologically the phase portraits of the systems we obtained, we use the following geometric invariants: • The number IS R of real infinite singularities.
• The number FS R of real finite singularities.
• The number Sep f of separatrices beginning or ending at a finite singularity.
• The number Sep ∞ of separatrices beginning or ending at an infinite singularity.
• The number FSep of separatrices connecting finite singularities.
• The number SC of separatrix connections.
• The maximum number ES ∞ of elliptic sectors in the vicinity of an infinite singularity.

Proof. (A)
The expressions for the integrating factors and the first integrals presented in Table 4.1 (see column 3) follow, after some easy calculations, by using Theorems 3.1 and 1.4.

(B)
We split the proof of this statement in three parts in accordance with the following three groups of the configurations (see Figure 3.1) : (α) Configurations 8.1-8.17:x = 2 the corresponding phase portraits are constructed in [36]; (β) Configurations 8. 18-8.23, 26, 31, 32, 33, 36, 38, 42, 47-51: the corresponding canonical systems do not depend on parameters;  As it is proved in the article [36], each one of the configurations (I.1)-(I.17)leads to a single phase portrait with the exception of configuration (I.9), which leads to 2 topologically distinct phase portraits.We observe that (I.9) corresponds to Config.8.6.So we conclude that the affine invariant conditions provided by Theorem 3.1 for the realization of configuration Config.8.j for j = 1, . . ., 5, 7, . . ., 17 guarantees the realization of the corresponding phase portrait P. 8.j, too.It remains to examine the two phase portraits given by Config.8.6 and to determine the necessary and sufficient conditions for their realization.
According to [36] a cubic system possessing the configuration (I.9) could be brought via an affine transformation and time rescaling to the form As it was proved in [36] these systems have the phase portrait given by Figure 1.9a if a < 1 and by Figure 1.9b if a > 1 in [36].
(β) We point out that each one of the 20 configurations Config.8. 18-8.23, 26, 31, 32, 33, 36, 38, 41-43, 47-51 corresponds to a system without parameters.The phase portraits of polynomial differential equations are usually presented in the Poincaré disc using the so called Poincaré compactification, see for details Chapter 5 of [15].The existence of the 8 invariant straight lines taking into account their multiplicities and the knowledge of the real elementary first integrals allows us to draw the 18 phase portraits corresponding to the above mentioned configurations as presented in Figure 4.1.
We note that the study of the phase portraits of systems without parameters can also be done using the algebraic program P4, see for details Chapters 9 and 10 of [15].
(γ) Among the 34 Config.(i) detect the finite real singularities of the systems and their types; (ii) examine if the information about the invariant straight lines and the types of finite singularities as well as the types of simple infinite singularities determine univocally the behavior of the trajectories at infinity; (ii) construct the corresponding phase portraits on the Poincaré disk.
However first we prove the next lemma.
Lemma 4.5.The one-parameter families of cubic systems which correspond to the configurations Config.
Examining the Jacobian matrix corresponding to the simple singularity N 1 (1, 0, 0) for each one of the above families of systems we detect that it equals 1 0 0 1 .Therefore this singularity for any of systems (4.4)-(4.6) is a star node and this completes the proof of the lemma.We observe that the invariant straight lines r + 2x + x 2 = 0 of systems (4.4) are real if Discrim[r + 2x + x 2 , x] = 4(1 − r) > 0 and they are complex if 1 − r < 0 (these invariant lines could not coincide due to r − 1 = 0).So in what follows we examine these two cases.
These systems possess the following invariant affine lines: In addition the line at infinity Z = 0 is a double one.All these invariant lines are distinct due to the condition 0 < b = 1 and by the relation L 1 < L 2 < L 3 we mean that the invariant line L 2 is located between invariant lines other two parallel lines L 1 and L 3 .We detect that systems (4.7) possess three finite singularities: M 1 (0, 0), M 2,3 (−1 ∓ b, 0).For a singularity M i (x i , y i ) we denote by λ Considering the conditions 0 < b = 1 we conclude that the singularity M 1 (0, 0) is a star node and M 2 (−1 − b, 0) is a saddle.On the other hand we observe that the singular point M 3 (−1 + b, 0) is a (stable) node if 0 < b < 1 and it is a saddle if b > 1.
We observe that the singular point M 1 (respectively M 2 ; M 3 ) is located at the intersection of invariant line y = 0 with L 1 (respectively L 2 ; L 3 ).
If 0 < b < 1 then the singular point M 3 (located on the line L 3 ) is a node.Taking into account that in this case we have L 2 < L 3 < L 1 and the fact that by Lemma 4.5 the infinite singular point N 1 (1, 0, 0) is a node, we obtain the phase portrait given by P. 8.25.
Assume now b > 1.In this case the singularity M 3 is a saddle and we have L 2 < L 1 < L 3 .Since N 1 (1, 0, 0) is a node we arrive in this case at the phase portrait given by P. 8.24.(4.9)

2)
These systems possess the unique real finite singular point M 1 (0, 0) which is located at the intersection of the (unique real) invariant lines x = 0 and y = 0.For this real finite singularity we determine λ 1 = 1 + b 2 = λ 2 , i.e. systems (4.9) possess a star node.
As by Lemma 4.5 the infinite singular point N 1 (1, 0, 0) is a star node it is clear that the unique possible phase portrait in this case corresponds to P. 8.27 (see Figure 4.1).In addition the line at infinity Z = 0 is a double one.All these invariant lines are distinct due to the condition 0 < b = 1.
On the other hand systems (4.10) possess three finite singularities, located at the intersections these invariant lines: M 1 (0, 0) and M 2,3 (1 ∓ b, 0).For these singularities we obtain λ (1) It is clear that due to the condition 0 < b = 1 the types of these singularities are well determined, and namely: the singularity M 1 (0, 0) is a saddle whereas M 2 (1 − b, 0) and M 3 (1 + b, 0) are both star nodes.On the other hand the position of the invariant lines (and consequently of these singularities) depends on the value of the parameter b.More exactly we have L 1 < L 2 < L 3 if 0 < b < 1 and L 2 < L 1 < L 3 if b > 1.As a result, considering that the infinite singularity N 1 (1, 0, 0) is a node (see Lemma 4.5) we arrive at the phase portrait given by P. 8.28 if b > 1 and by P. 8.29 if 0 < b < 1.
2 (4.12) These systems possess the unique real finite singular point M 1 (0, 0) which is located at the intersection of the (unique real) invariant lines x = 0 and y = 0.For this real finite singularity we determine λ 1 = 1 + b 2 and λ 2 = −2(1 + b 2 ), i.e. systems (4.12) possess a saddle at the origin of coordinate.
As by Lemma 4.5 the infinite singular point N 1 (1, 0, 0) is a star node, clearly we obtain the unique possible phase portrait in this case and it corresponds to P. 8.30 (see Figure 4.1).

1 )
The case 1 − r > 0. We set 1 − r = b 2 > 0 where b is a new parameter and we get: r = 1 − b 2 < 1.We may consider b > 0 due to the change b → −b and we arrive at the family of systems ẋ