Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 50, pp. 1–38. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.50 ABEL QUADRATIC DIFFERENTIAL SYSTEMS OF SECOND KIND JOAN C. ARTÉS, JAUME LLIBRE, DANA SCHLOMIUK, NICOLAE VULPE Abstract. The Abel differential equations of second kind, named after Niels Henrik Abel, are a class of ordinary differential equations studied by many authors. Here we consider the Abel quadratic polynomial differential equations of second kind denoting this class by QSAb. Firstly we split the whole family of non-degenerate quadratic systems in four subfamilies according to the number of infinite singularities. Secondly for each one of these four subfamilies we determine necessary and sufficient affine invariant conditions for a quadratic system in this subfamily to belong to the class QSAb. Thirdly we classify all the phase portraits in the Poincaré disc of the systems in QSAb in the case when they have at infinity either one triple singularity (21 phase portraits) or an infinite number of singularities (9 phase portraits). Moreover we determine the affine invariant criteria for the realization of each one of the 30 topologically distinct phase portraits. 1. Introduction and statement of main results We consider the class of real quadratic polynomial differential systems ẋ = p0 + p1(x, y) + p2(x, y) ≡ P (ã, x, y), ẏ = q0 + q1(x, y) + q2(x, y) ≡ Q(ã, x, y) (1.1) where p0 = a, p1(x, y) = cx+ dy, p2(x, y) = gx2 + 2hxy + ky2, q0 = b, q1(x, y) = ex+ fy, q2(x, y) = lx2 + 2mxy + ny2 and with max(deg(p),deg(q)) = 2. Here the dot denotes derivative with respect to an independent variable t, usually called the time. We denote by ã = (a, c, d, g, h, k, b, e, f, l,m, n) the 12-tuple of the coefficients of systems (1.1), and by QS the class of all real quadratic polynomial differential systems, sometimes are simply called quadratic systems. There are more than one thousand papers published on QS (see for instance [15]). The main difficulty of studying QS comes from the fact that they depend on twelve parameters. However modulo the affine group action and time rescaling 2020 Mathematics Subject Classification. 58K45, 34C23, 34A34. Key words and phrases. Quadratic differential systems; phase portraits; second kind of Abel differential equations; affine invariant polynomials. ©2024. This work is licensed under a CC BY 4.0 license. Submitted July 10, 2024. Published September 4, 2024. 1 2 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 this family depends on five parameters, still a large number. So people began by studying subclasses of QS depending on at most three parameters. For a nice extensive survey on quadratic differential systems, and an extended bibliography on them, the reader is advised to consult [5, Chapter 2]. In this article we study the Abel differential equations of second kind which are of the form y dy dx = A(x)y2 +B(x)y + C(x), (1.2) where A(x), B(x) and C(x) are rational functions over R. The above equations are equivalent to the differential systems ẋ = d(x)y, ẏ = a(x)y2 + b(x)y + c(x), where A(x) = a(x)/d(x), B(x) = b(x)/d(x) and C(x) = c(x)/d(x). In this article we are interested in studying the Abel quadratic polynomial differ- ential systems, i.e. differential systems of the form ẋ = (d0+d1x)y ≡ P̃ (x, y), ẏ = a0y 2+(b0+b1x)y+c0+c1x+c2x 2 ≡ Q̃(x, y), (1.3) coming from the Abel differential equation of second kind (1.2). Definition 1.1. We say that a non-degenerate quadratic system (1.1) is of Abel type if there exists an affine transformation which brings this system to the form (1.3). We denote the class of quadratic systems of Abel type by QSAb . Some subclasses of QSAb have already been studied. The family of systems (1.3) with a0 = 0 and with Z2-symmetries is considered in [13]. The family of systems (1.3) with d1 = 0 and a0 ̸= 0 is analyzed in [10]. Finally, in [11] the family of systems (1.3) with a0 ̸= 0 and having a symmetry with respect to an axis or with respect to the origin is considered. The goal of this article is firstly to determine necessary and sufficient conditions in terms of affine invariant polynomials for an arbitrary quadratic system to be of Abel type. Secondly, we classify topologically all the phase portraits in the Poincaré disc of the systems in QSAb in the case when they have at infinity either one triple singularity (when all three infinite singularities coalesce), or an infinite number of singularities. Moreover we want to determine the affine invariant criteria for the realization of each one of the 30 topologically distinct phase portraits. The affine invariant polynomials which appear in the statement of the next theorem are defined in Section 2. Our main result is the following one. Theorem 1.2. A non-degenerate quadratic system (1.1) (i.e. ∑4 i=0 µ 2 i ̸= 0) belongs to the class QSAb of Abel quadratic systems if and only if B1 = 0 and one of the following conditions is satisfied: (A) If η > 0 then either (A1) θ ̸= 0, or (A2) θ = 0, Ñ ̸= 0, H7 ̸= 0, or (A3) θ = 0, Ñ ̸= 0, H7 = 0, B2 = 0, or (A4) θ = 0, Ñ = 0, θ3 ̸= 0, or (A5) θ = 0, Ñ = 0, θ3 = 0, B2 = 0, θ4 ̸= 0, or (A6) θ = 0, Ñ = 0, θ3 = 0, B2 = 0, θ4 = 0, B3 = 0. (B) If η < 0 then either (B1) θ ̸= 0, B2 ̸= 0, or EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 3 (B2) θ ̸= 0, B2 = 0, B3 = 0, or (B3) θ = 0, Ñ ̸= 0, H7 ̸= 0, B2 ̸= 0, or (B4) θ = 0, Ñ ̸= 0, H7 ̸= 0, B2 = 0, B3 = 0, or (B5) θ = 0, Ñ = 0, B2 ̸= 0, or (B6) θ = 0, Ñ = 0, B2 = 0, B3 = 0. (C) If η = 0 and M̃ ̸= 0 then either (C1) θ ̸= 0, or (C2) θ = 0, µ0 ̸= 0, H7 ̸= 0, or (C3) θ = 0, µ0 ̸= 0, H7 = 0, B2 = 0, or (C4) θ = 0, µ0 = 0, Ñ ̸= 0, H7 ̸= 0, or (C5) θ = 0, µ0 = 0, Ñ ̸= 0, H7 = 0, B3 = 0, or (C6) θ = 0, µ0 = 0, Ñ = 0, K̃ ̸= 0, θ3 ̸= 0, or (C7) θ = 0, µ0 = 0, Ñ = 0, K̃ ̸= 0, θ3 = 0, B3 = 0. (D) If η = 0 and M̃ = 0 then either (D1) C2 ̸= 0, θ ̸= 0, or (D2) C2 ̸= 0, θ = 0, Ñ = 0, B2 ̸= 0, or (D3) C2 = 0, H10 ̸= 0, or (D4) C2 = 0, H10 = 0, H12 ̸= 0. The phase portraits of the subcase (D) will be obtained in Theorems 4.1 and 4.11 in Section 4.4. We are able to find all the phase portraits of subcase (D) because they are the less generic ones, and use less parameters. The other subcases are more generic and much more difficult to study. We have 30 phase portraits, none with limit cycles but 10 with graphics surrounding a focus or a center. 2. Main invariant polynomials associated with the class QSAb Consider quadratic systems of the form (1.1). It is known that on the set QS acts the group Aff(2,R) of affine transformations on the plane (cf. [17]). For every subgroup G ⊆ Aff(2,R) we have an induced action of G on QS . We can identify the set QS of systems (1.1) with a subset of R12 via the map QS −→ R12 which associates to each system (1.1) the 12–tuple ã = (a, c, d, g, h, k, b, e, f, l,m, n) of its coefficients. We associate to this group action polynomials in x, y and parameters which behave well with respect to this action, the GL–comitants (GL–invariants), the T–comitants (affine invariants) and the CT–comitants. For their definitions as well as their detailed constructions we refer the reader to the paper [17] (see also [5]). Next we define the following 41 invariant polynomials associated with the class QSAb: { µ0, . . . , µ4, D, P, R, S, T, U, T1, . . . , T4, F , F1, . . . ,F4, H, B, B1, B2, σ, η, M̃ , C2, θ, θ3, θ4, K̃, Ñ , H7, H9, H10, H11, H12, E1, U1, U2 } . (2.1) 4 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 According to [5] (see also [7]) we apply the differential operator L = x ·L2−y ·L1 acting on R[ã, x, y] with L1 =2a ∂ ∂c + c ∂ ∂g + 1 2 d ∂ ∂h + 2b ∂ ∂e + e ∂ ∂l + 1 2 f ∂ ∂m , L2 =2a ∂ ∂d + d ∂ ∂k + 1 2 c ∂ ∂h + 2b ∂ ∂f + f ∂ ∂n + 1 2 e ∂ ∂m , to construct several invariant polynomials from the set. More precisely using this operator and the affine invariant µ0 = Resx ( p2(ã, x, y), q2(ã, x, y) ) /y4 we construct the following polynomials µi(ã, x, y) = 1 i! L(i)(µ0), i = 1, . . . , 4, where L(i)(µ0) = L(L(i−1)(µ0)). Using these invariant polynomials we define some new ones, which according to [5] are responsible for the number and multiplicities of the finite singular points of (1.1): D = [ 3 ( (µ3, µ3) (2), µ2 )(2) − ( 6µ0µ4 − 3µ1µ3 + µ2 2, µ4 )(4)] /48, P = 12µ0µ4 − 3µ1µ3 + µ2 2, R = 3µ2 1 − 8µ0µ2, S = R2 − 16µ2 0P, T = 18µ2 0(3µ 2 3 − 8µ2µ4) + 2µ0(2µ 3 2 − 9µ1µ2µ3 + 27µ2 1µ4)−PR, U = µ2 3 − 4µ2µ4. In what follows we also need the so-called transvectant of order k (see [12, 14] of two polynomials f , g ∈ R[ã, x, y] (f, g)(k) = k∑ h=0 (−1)h ( k h ) ∂kf ∂xk−h∂yh ∂kg ∂xh∂yk−h . Following [22] we denote by σ(ã, x, y) = ∂P ∂x + ∂Q ∂y = σ0(ã) + σ1(ã, x, y) and we observe that the polynomial σ(ã, x, y) ∈ R[x, y] is an affine comitant of systems (1.1). Next we construct the elements T1, . . . , T4 of the set (2.1) which are responsible for the number of the vanishing traces corresponding to the finite singularities of systems (1.1). For this we define a polynomial (which we call trace polynomial) as follows. Definition 2.1 ([22]). We call trace polynomial T(w) over the ring R[ã] the poly- nomial defined as follows T(w) = 4∑ i=0 1 (i!)2 ( σi 1, 1 i! L(i)(µ0) )(i) w4−i = 4∑ i=0 Gi(ã)w 4−i, where the coefficients Gi(ã) = 1 (i!)2 (σi 1, µi) (i) ∈ R[ã], i = 0, 1, 2, 3, 4 ( G0(ã) ≡ µ0(ã) ) are GL-invariants. EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 5 Using the polynomial T(w) we could construct the above mentioned four affine invariants T4, T3, T2 and T1: T4−i(ã) = 1 i! diT dwi ∣∣∣ w=σ0 ∈ R[ã], i = 0, 1, 2, 3 ( T4 ≡ T(σ0) ) . To construct the remaining invariant polynomials contained in the set (2.1) we first need to define some elementary bricks which help to construct these elements of the set. We remark that the following polynomials in R[ã, x, y] are the simplest invariant polynomials of degree one with respect to the coefficients of the differential systems (1.1) and which are GL-comitants: Ci(x, y) = ypi(x, y)− xqi(x, y), i = 0, 1, 2; Di(x, y) = ∂ ∂x pi(x, y) + ∂ ∂y qi(x, y), for i = 1, 2. Apart from these simple invariant polynomials we shall also make use of the following nine GL-invariant polynomials: T1 = (C0, C1) (1) , T2 = (C0, C2) (1) , T3 = (C0, D2) (1) , T4 = (C1, C1) (2) , T5 = (C1, C2) (1) , T6 = (C1, C2) (2) , T7 = (C1, D2) (1) , T8 = (C2, C2) (2) , T9 = (C2, D2) (1) . These are of degree two with respect to the coefficients of systems (1.1). We next define a list of T -comitants: Â(ã) = (C1, T8 − 2T9 +D2 2) (2)/144, B̂(ã, x, y) = { 16D1(D2, T8) (1)(3C1D1 − 2C0D2 + 4T2) + 32C0(D2, T9) (1)(3D1D2 − 5T6 + 9T7) + 2(D2, T9) (1) ( 27C1T4 − 18C1D 2 1 − 32D1T2 + 32(C0, T5) (1) ) + 6(D2, T7) (1) [ 8C0(T8 − 12T9)− 12C1(D1D2 + T7) +D1(26C2D1 + 32T5) + C2(9T4 + 96T3) ] + 6(D2, T6) (1) [ 32C0T9 − C1(12T7 + 52D1D2) − 32C2D 2 1 ] + 48D2(D2, T1) (1)(2D2 2 − T8) + 6D1D2T4(T8 − 7D2 2 − 42T9) − 32D1T8(D2, T2) (1) + 9D2 2T4(T6 − 2T7)− 16D1(C2, T8) (1)(D2 1 + 4T3) + 12D1(C1, T8) (2)(C1D2 − 2C2D1) + 12D1(C1, T8) (1)(T7 + 2D1D2) + 96D2 2 [ D1(C1, T6t) (1) +D2(C0, T6) (1) ] − 4D3 1D2(D 2 2 + 3T8 + 6T9) − 16D1D2T3(2D 2 2 + 3T8) + 6D2 1D 2 2(7T6 + 2T7)− 252D1D2T4T9 } /(2833), D̂(ã, x, y) = [ 2C0(T8 − 8T9 − 2D2 2) + C1(6T7 − T6)− (C1, T5) (1) − 9D2 1C2 + 6D1(C1D2 − T5) ] /36, Ê(ã, x, y) = [ D1(2T9 − T8)− 3(C1, T9) (1) −D2(3T7 +D1D2) ] /72, F̂ (ã, x, y) = [ 6D2 1(D 2 2 − 4T9) + 4D1D2(T6 + 6T7) + 48C0(D2, T9) (1) − 9D2 2T4 + 288D1Ê − 24(C2, D̂)(2) + 120(D2, D̂)(1) − 36C1(D2, T7) (1) + 8D1(D2, T5) (1) ] /144, 6 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 K̂(ã, x, y) = (T8 + 4T9 + 4D2 2)/72, Ĥ(ã, x, y) = (−T8 + 8T9 + 2D2 2)/72, as well as the needed bricks: A1(ã) = Â, A2(ã) = (C2, D̂)(3)/12, A3(ã) = [[C2, D2) (1), D2 )(1) , D2 )(1) /48, A4(ã) = (Ĥ, Ĥ)(2), A5(ã) = (Ĥ, K̂)(2)/2, A6(ã) = (Ê, Ĥ)(2)/2, A7(ã) = [[C2, Ê)(2), D2 )(1) /8, A8(ã) = [[D̂, Ĥ)(2), D2 )(1) /8, A9(ã) = [[D̂,D2) (1), D2 )(1) , D2 )(1) /48, A10(ã) = [[D̂, K̂)(2), D2 )(1) /8, A11(ã) = (F̂ , K̂)(2)/4, A12(ã) = (F̂ , Ĥ)(2)/4, A14(ã) = (B̂, C2) (3)/36, A15(ã) = (Ê, F̂ )(2)/4, A25(ã) = [[D̂, D̂)(2), Ê )(2) /16, plus A33(ã) = [[D̂,D2) (1), F̂ )(1) , D2 )(1) , D2 )(1) /128, A34(ã) = [[D̂, D̂)(2), D2 )(1) , K̂ )(1) , D2 )(1) /64. In the above list the bracket “[[” means a succession of two or up to four paren- theses “(” depending on the row where they appear. Now we can define the remaining invariant polynomials of the set (2.1): F(ã) = A7, F1(ã) =, A2, F2(ã) = −2A2 1A3 + 2A5(5A8 + 3A9) +A3(A8 − 3A10 + 3A11 +A12) −A4(10A8 − 3A9 + 5A10 + 5A11 + 5A12), F3(ã) = −10A2 1A3 + 2A5(A8 −A9)−A4(2A8 +A9 +A10 +A11 +A12) +A3(5A8 +A10 −A11 + 5A12), F4(ã) = 20A2 1A2 −A2(7A8 − 4A9 +A10 +A11 + 7A12) +A1(6A14 − 22A15)− 4A33 + 4A34, H(ã) = −(A4 + 2A5), B(ã) = −(3A8 + 2A9 +A10 +A11 +A12), B1(ã, x, y) = {( T7, D2 )(1)[ 12D1T3 + 2D3 1 + 9D1T4 + 36 ( T1, D2 )(1)] − 2D1 ( T6, D2 )(1) (D2 1 + 12T3) +D2 1 [ D1 ( T8, C1 )(2) + 6 (( T6, C1 )(1) , D2 )(1)]} /144, B2(ã, x, y) = {( T7, D2 )(1)[ 8T3 ( T6, D2 )(1) −D2 1 ( T8, C1 )(2) − 4D1 (( T6, C1 )(1) , D2 )(1)] + [( T7, D2 )(1)]2 (8T3 − 3T4 + 2D2 1) } /384, K̃(ã, x, y) = 4K̂ ≡ Jacob ( p2(ã, x, y), q2(ã, x, y) ) , M̃(ã, x, y) = (C2, C2) (2) ≡ 2Hess ( C2(ã, x, y) ) , EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 7 Ñ(ã, x, y) = K̃ − 4Ĥ, η(ã) = (M̃, M̃)(2)/384 ≡ Discrim ( C2(ã, x, y) ) , θ(ã) = −(Ñ , Ñ)(2)/2 ≡ Discrim ( Ñ(ã, x, y) ) ; θ3(ã) = A8 +A11, θ4(ã) = A7, B1(ã) = Resx ( C2, D̃ ) /y9 = −2−93−8 (B2, B3) (4) , B2(ã, x, y) = (B3, B3) (2) − 6B3(C2, D̃)(3), B3(ã, x, y) = (C2, D̃)(1) ≡ Jacob ( C2, D̃ ) , E1(ã) = A25, Ũ1(ã) = A9 − 54A2 1, Ũ2(ã) = 3A8 −A9, H7(ã) = (Ñ , C1) (2), H9(ã) = −[[D̃, D̃)(2), D̃, )(1) , D̃ )(3) , H10(ã) = [[Ñ , D̃)(2), D2 )(1) , H11(ã, x, y) = 8Ĥ [ (C2, D̃)(2) + 8(D̃,D2) (1) ] + 3 [ (C1, 2Ĥ − Ñ)(1) − 2D1Ñ ]2 , H12(ã, x, y) = (D̃, D̃)(2) ≡ Hessian(D̃). We remark that the above invariant polynomials (except Ũ1 and Ũ2) were con- structed and used in [22], [18] and [3], and only the invariant polynomials Ũ1 and Ũ2 are defined here. 3. Preliminary results involving the use of polynomial invariants We remark that the invariant polynomials µi(ã, x, y) (i = 0, 1, . . . , 4) defined in the previous subsection are responsible for the total multiplicity of the finite singularities of quadratic systems (1.1). Moreover they detect whether a quadratic system is degenerate or not. More exactly we have the following lemma. Lemma 3.1 ([5, Lemma 5.2]). Consider a system (S) from the family (1.1) with coefficients ã ∈ R12. Then: (i) The total multiplicity of the finite singularities of this system is 4−k if and only if for every i such that 0 ≤ i ≤ k− 1 we have µi(ã, x, y) = 0 in R[x, y] and µk(ã, x, y) ̸= 0. (i) The system (S) is degenerate (i.e. gcd(P,Q) ̸= const) if and only if µi(ã, x, y) = 0 in R[x, y] for every i = 0, 1, 2, 3, 4. On the other hand the invariant polynomials η, M̃ and C2 govern the number of real and complex infinite singularities. More precisely, according to [19] (see also [17]) we have the next result. Lemma 3.2. The number of infinite singularities (real and complex) of a quadratic system in QS is determined by the following conditions: (i) 3 real if η > 0; (ii) 1 real and 2 imaginary if η < 0; (iii) 2 real if η = 0 and M̃ ̸= 0; (iv) 1 real if η = M̃ = 0 and C2 ̸= 0; (v) ∞ if η = M̃ = C2 = 0. 8 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 Moreover, the quadratic systems (1.1), for each one of these cases, can be brought via a linear transformation to the corresponding case of the following canonical systems (SI)− (SV ):{ ẋ = a+ cx+ dy + gx2 + (h− 1)xy, ẏ = b+ ex+ fy + (g − 1)xy + hy2; (SI){ ẋ = a+ cx+ dy + gx2 + (h+ 1)xy, ẏ = b+ ex+ fy − x2 + gxy + hy2; (SII){ ẋ = a+ cx+ dy + gx2 + hxy, ẏ = b+ ex+ fy + (g − 1)xy + hy2; (SIII) { ẋ = a+ cx+ dy + gx2 + hxy, ẏ = b+ ex+ fy − x2 + gxy + hy2; (SIV ){ ẋ = a+ cx+ dy + x2, ẏ = b+ ex+ fy + xy. (SV ) According to [4] (see also [5]) the next proposition holds. Proposition 3.3. Consider a non-degenerate quadratic differential system. Then: (i) this system has one center if and only if one of the following sets of conditions holds (C1) T4 = 0, T3F < 0, F1 = F2 = F3F4 = 0; (C2) T4 = T3 = 0, T2 > 0, B < 0, F = F1 = 0; (C3) T4 = T3 = T2 = T1 = 0, σ ̸= 0, F1 = 0,H < 0, B < 0, F = 0; (C4) T4 = T3 = T2 = T1 = 0, σ ̸= 0, F1 = 0,H = B1 = 0, B2 < 0; (C5) σ = 0, µ0 < 0, D < 0, R > 0, S > 0; (C6) σ = 0, µ0 = 0, D < 0, R ̸= 0; (C7) σ = 0, µ0 > 0, D > 0; (C8) σ = 0, µ0 > 0, D = 0, T < 0; (C9) σ = 0, µ0 = µ1 = 0, µ2 ̸= 0,U > 0, K̃ = 0; (C10) σ = 0, µ0 > 0,D = T = P = 0, R ̸= 0; (3.1) (ii) and it has two centers if and only if one of the following sets of conditions holds (Ĉ1) T4 = T3 = 0, T2 < 0, B < 0, H < 0, F = F1 = 0; (Ĉ2) σ = 0, µ0 > 0, D < 0, R > 0, S > 0. (3.2) In what follows we also need the next lemma. Lemma 3.4 ([16]). For the existence of invariant straight lines of a system (1.1) in one (respectively 2; 3 distinct) directions in the affine plane it is necessary that B1 = 0 (respectively B2 = 0; B3 = 0). 4. Proof of Theorem 1.2 We shall consider step by step each one of the subfamilies of quadratic systems defined by the conditions (A) - (D) which are provided by Theorem 1.2. EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 9 4.1. Subfamily defined by (A): η > 0. According to Lemma 3.2 we consider systems (SI) for which calculations yield: η = 1, θ = −8(g − 1)(h− 1)(g + h). We examine two cases: θ ̸= 0 and θ = 0. 4.1.1. Case θ ̸= 0. Then (g− 1)(h− 1)(g+h) ̸= 0 and due to a translation we may assume d = e = 0, i.e. we obtain the systems ẋ = a+ cx+ gx2 + (h− 1)xy, ẏ = b+ fy + (g − 1)xy + hy2, (4.1) for which we calculate B1 = ab(g − 1)2(h− 1)2 [ (b− a)(g + h)2 + cf(g − h) + c2h− f2g ] ≡ ab(g − 1)2(h− 1)2H. So because θ ̸= 0 the condition B1 = 0 is equivalent to abH = 0 and we consider two subcases: ab = 0 and ab ̸= 0. 1: ab = 0. We observe that systems (4.1) keep the form under the change (x, y, a, b, c, f, g, h) 7→ (y, x, b, a, f, c, h, g), and hence without loss of generality we may consider that the condition a = 0 is fulfilled. Then we arrive at the family of systems ẋ = x [ c+ gx+ (h− 1)y ] , ẏ = b+ fy + (g − 1)xy + hy2, possessing the invariant affine line x = 0. It is not too difficult to see, that after the affine transformation x1 = x, y1 = gx+ (h− 1)y + c, we arrive at the systems ẋ1 = x1y1, ẏ1 = b′ + e′x1 + l′x2 1 + (f ′ + 2m′x1)y1 + n′y2, (4.2) where b′, e′, f ′, l′,m′ and n′ are rational functions of the parameters b, c, f, g, h with the same denominator h − 1 ̸= 0. We observe that these systems belong to the family of systems (1.3). 2: ab ̸= 0. In this case we get H = 0. Then from the equality (b− a)(g + h)2 + cf(g − h) + c2h− f2g = 0, we obtain b = a+ (f − c)(fg + ch) (g + h)2 ≡ b0, and this leads to the family of systems ẋ = a+ cx+ gx2 + (h− 1)xy, ẏ = b0 + fy + (g − 1)xy + hy2. Since g + h ̸= 0 we can apply to these systems the transformation x1 = (g + h)(x− y) + c− f, y1 = (g + h)(gx+ hy) + fg + ch, t1 = t/(g + h) with the determinant (g + h)3 ̸= 0. Then we obtain the family of systems (4.2), where the parameters b′, e′, f ′, l′,m′ and n′ are rational functions of the parameters a, c, f, g, h with the same denominator g+h ̸= 0. So we again arrive at a subfamily of systems (1.3). 10 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 4.1.2. Case θ = 0. For systems (SI) we have θ = −8(g−1)(h−1)(g+h), Ñ = (g2−1)x2+2(g−1)(h−1)xy+(h2−1)y2, (4.3) and therefore the condition θ = 0 yields (h − 1)(g − 1)(g + h) = 0. Without loss of generality we can consider h = 1. Indeed, if g = 1 (respectively, g + h = 0) we can apply the linear transformation that replaces the straight line x = 0 by y = 0 (respectively, x = 0 by y = x) reducing this case to h = 1. So we assume h = 1 and in this case by (4.3) for systems (SI) we have Ñ = (g − 1)(1 + g)x2. We consider two subcases: Ñ ̸= 0 and Ñ = 0. 1: Ñ ̸= 0. Then (g − 1)(g + 1) ̸= 0 and due to a translation we may assume e = f = 0. So we obtain the family of systems ẋ = a+ cx+ dy + gx2, ẏ = b+ (g − 1)xy + y2, (4.4) for which we calculate B1 = bd2g(g − 1)2 [ (b− a)(1 + g)2 + (c+ d)(c− dg) ] ≡ bd2g(g − 1)2Φ, µ0 = g2, H7 = 4d(g2 − 1). 1.1: H7 ̸= 0. This implies d(g− 1) ̸= 0 and therefore the condition B1 = 0 yields bgΦ = 0. We consider two cases: µ0 ̸= 0 and µ0 = 0. 1.1.1: µ0 ̸= 0. Then g ̸= 0 and we obtain bΦ = 0. 1.1.1.1: b = 0. Then systems (4.4) possess the invariant line y = 0 and using the transformation x1 = y, y1 = (g − 1)x+ y, we arrive at the systems ẋ1 = x1y1, ẏ1 = a(g− 1)− (c+ d− dg)x1 + cy1 + 1 g − 1 [ gx2 1 − (g− 1)x1y1 + gy21 ] . So we obtain a subfamily of the family of systems (1.3). 1.1.1.2: Φ = 0. This condition gives b = a(1 + g)2 − (c+ d)(c− dg) (1 + g)2 ≡ b0, and systems (4.4) with b = b0 possess the invariant line (1 + g)(x− y) + c+ d = 0. Then applying the transformation x1 = (1 + g)(x− y) + c+ d, y1 = gx+ y (Det = (1 + g)2 ̸= 0), we obtain a subfamily of Abel quadratic systems of the form (1.3): ẋ1 = x1y1, ẏ1 = Q(x1, y1), where Q(x1, y1) is a quadratic polynomial whose coefficients are rational functions of the parameters a, c, d, g, h with the denominators being some powers of g+1 ̸= 0. 1.1.1: µ0 = 0. Then we have g = 0 and considering systems (4.4) we obtain the systems ẋ = a+ cx+ dy, ẏ = b− xy + y2. EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 11 Since d ̸= 0 we can apply the transformation x1 = x, y1 = cx+ dy + a, which brings the above systems to the form ẋ1 = y1, ẏ1 = 1 d [ a2+bd2+a(2c+d)x1+(cd−2a)y1+c(c+d)x2 1−(2c+d)x1y1+y21 ] . It is clear that these systems are contained in the family of systems (1.3) (in the first equation we have d1 = 0 and d0 = 1). 1.2: H7 = 0. Then d = 0 and we arrive at the family of systems ẋ = a+ cx+ gx2, ẏ = b+ (g − 1)xy + y2, (4.5) for which we calculate B1 = 0, B2 = −648b(−1 + g)2 [ (b− a)(1 + g)2 + c2 ] x4, µ0 = g2 and we consider two cases: B2 ̸= 0 and B2 = 0. 1.2.1: B2 ̸= 0. We claim that for B2 ̸= 0, systems (4.5) could not be brought via an affine transformation to the form (1.3). To prove this claim we examine two subcases: µ0 ̸= 0 and µ0 = 0. 1.2.1.1: µ0 ̸= 0. Then g ̸= 0 and systems (4.5) possess two parallel invariant lines a+ cx+ gx2 = 0 (which can be real or complex or coinciding). On the other hand for systems (1.3) we have µ0 = a0c2d 2 1 and the condition µ0 ̸= 0 implies d1 ̸= 0. This means that systems (1.3) possess invariant line d0+d1x = 0 and there does not exist another parallel invariant line in the direction x = 0. It remains to observe that according to Lemma 3.4 for the existence of invariant lines in two distinct directions for a quadratic system the condition B2 = 0 is necessary. Therefore systems (4.5) for B2 ̸= 0 could not have an invariant affine line in other direction, which could be used for the construction of the needed affine transformation. 1.2.1.2: µ0 = 0. Then g = 0 and considering (4.5) we obtain the systems ẋ = a+ cx, ẏ = b− xy + y2, (4.6) for which we have B1 = µ0 = H7 = 0, Ñ = −x2. On the other hand for systems (1.3) we have µ0 = a0c2d 2 1 and the condition µ0 = 0 gives a0c2d1 = 0. If d1 = 0 then for systems (1.3) we calculate H7 = −4(b21 − 4a0c2)d0, Ñ = −(b21 − 4a0c2)x 2 and the conditions H7 = 0 and Ñ ̸= 0 imply d0 = 0 which leads to degenerate systems (1.3). Assume now d1 ̸= 0. This means that systems (1.3) possess invariant line d0 + d1x = 0 in the direction x = 0 and therefore (d0 + d1x) is a factor in P̃ (x, y). Moreover the second factor of P̃ (x, y) in (1.3) is y. On the other hand systems (4.6) could possess in the direction x = 0 either one invariant affine line a + cx = 0 if c ̸= 0 or zero lines if c = 0. Moreover the right hand side of the first equation does not contain the factor y. 12 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 It remains to observe that according to Lemma 3.4 systems (4.6) for B2 ̸= 0 could not have an invariant affine line in another direction, which could be used for the construction of the needed affine transformation. This completes the proof of the claim. 1.1.2: B2 = 0. Then b [ (b − a)(1 + g)2 + c2 ] = 0. We observe that the second factor equals Φ ∣∣ d=0 and we deduce that we can apply the same arguments as previously in the case H7 ̸= 0 repeating the steps 1.1.1.1 (b = 0) and 1.1.1.2 (Φ = 0) and considering the condition d = 0. Thus the condition B2 = 0 guarantees the existence of an affine transformation which brings systems (4.5) to the form (1.3). 2: Ñ = 0. Considering (4.3) the condition Ñ = 0 yields (g − 1)(h− 1) = g2 − 1 = h2 − 1 = 0 and we obtain three possibilities: (a) g = 1 = h; (b) g = 1 = −h; (c) g = −1 = −h. The cases (b) and (c) can be brought by linear transformations to the case (a). So g = h = 1 and systems (SI) after an additional translation (to make c = d = 0) are of the form: ẋ = a+ dy + x2, ẏ = b+ ex+ y2. (4.7) For these systems we calculate B1 = −d2e2(4a− 4b+ d2 − e2), µ0 = 1, θ3 = −2de, θ4 = −(d+ e), and we consider two possibilities: θ3 ̸= 0 and θ3 = 0. 2.1: θ3 ̸= 0. Then the condition B1 = 0 implies b = a + (d2 − e2)/4 and we obtain the systems ẋ = a+ dy + x2, ẏ = a+ (d2 − e2)/4 + ex+ y2, (4.8) possessing the invariant line 2x−2y+d−e = 0. So by means of the transformation x1 = 2x− 2y + d− e, y1 = x+ y − (d+ e)/2, we arrive at the following subfamily of (1.3): ẋ1 = x1y1, ẏ1 = (4a+ 2d2 + e2)/2 + (e− d)x1 + (d+ e)y1 + x2 1/8 + y21/2. (4.9) 2.2: θ3 = 0. Then de = 0 and we may consider d = 0 due to the change (x, y, a, b, d, e) 7→ (y, x, b, a, e, d) which conserves the systems. In this case we have B1 = 0, B2 = 648e2(4a− 4b− e2)x4, θ4 = −e, and we consider two cases: B2 ̸= 0 and B2 = 0. 2.2.1: B2 ̸= 0. We claim that for B2 ̸= 0 systems (4.7) with d = 0 could not be brought via an affine transformation to the form (1.3). Indeed, for systems (1.3) we calculate µ0 = a0c2d 2 1 ̸= 0 (since for (4.7) we have µ0 = 1). Hence d1 ̸= 0 and these systems possess a single invariant line d0+d1x = 0 in the direction x = 0. On the other hand systems (4.7) with d = 0 possess in the direction x = 0 two parallel invariant lines x2 + a = 0, which could be real or complex or coinciding. Taking into account that by Lemma 3.4 in the case B2 ̸= 0 these systems could not have invariant lines in other directions we conclude that the claim is proved. EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 13 2.2.2: B2 = 0. Then e(4a− 4b− e2) = 0 and we examine two subcases: θ4 ̸= 0 and θ4 = 0. 2.2.2.1: θ4 ̸= 0. In this case we get b = a− e2/4 and we obtain systems (4.8) with d = 0. So applying the transformation x1 = 2x− 2y − e, y1 = x+ y − e/2, we arrive at the family of systems (4.9) with d = 0 which is a subfamily of (1.3). 2.2.2.2: θ4 = 0. Then e = 0 and we obtain the systems ẋ = a+ x2, ẏ = b+ y2, (4.10) for which calculations yield B1 = B2 = 0, B3 = −12(a− b)x2y2, µ0 = 1. These systems have two couples of parallel lines: a + x2 = 0 (in the direction x = 0) and b+ y2 = 0 (in the direction y = 0) which could be real, or complex, or coinciding. We examine two possibilities: B3 ̸= 0 and B3 = 0. 2.2.2.2.1: B3 ̸= 0. Then by Lemma 3.4 systems (4.10) could not have other invariant lines. On the other hand, as it was mentioned earlier, since µ0 ̸= 0, systems (1.3) have a single invariant line in the direction x = 0. So we conclude that systems (4.10) could not be brought to the form (1.3) by means of an affine transformation. 2.2.2.2.2: B3 = 0. This implies b = a and then systems (4.10) possess also the invariant line y = x. So applying the transformation x1 = x− y, y1 = x+ y we obtain the family of systems ẋ1 = x1y1, ẏ1 = 2a+ x2 1/2 + y21/2, which is a subfamily of (1.3). As all the possibilities in the case η > 0 are examined we conclude that the statement (A) of Theorem 1.2 is proved. 4.2. Subfamily defined by (B): η < 0. In this case by Lemma 3.2 we have to consider the systems (SII) for which we calculate: η = −4, θ = 8(1 + h) [ g2 + (h− 1)2 ] , Ñ = (g2 − 2h+ 2)x2 + 2g(h+ 1)xy + (h2 − 1)y2. (4.11) So we examine two cases: θ ̸= 0 and θ = 0. 4.2.1. Case θ ̸= 0. Then h+1 ̸= 0 and by a translation we cany assume c = d = 0, i.e. we obtain the systems ẋ = a+ gx2 + (h+ 1)xy, ẏ = b+ ex+ fy − x2 + gxy + hy2. (4.12) For these systems we calculate B1 = −a(h+ 1)2(α2 + β2), B2 = −648 [ α(α+ a(1 + h)2) + β2 ] x4 + 648a(1 + h)2αy2(6x2 − y2) − 2592a(1 + h)2βxy(x2 − y2), where α = a [ g2 − (h− 1)2 ] − 2bg(h− 1) + f(−e+ fg − eh), β = 2ag(h− 1) + b [ g2 − (h− 1)2 ] − f2 − efg + e2h. (4.13) 14 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 It is not too difficult to observe that the condition α = β = 0 is equivalent to B1 = B2 = 0. So we consider two subcases: B2 ̸= 0 and B2 = 0. 1: B2 ̸= 0. Then by θ ̸= 0 the condition B1 = 0 implies a = 0 and applying the transformation x1 = x, y1 = gx+ (1 + h)y, we arrive at the family of systems ẋ1 = x1y1, ẏ1 = b(1 + h) + (e− fg + eh)x1 + fy1 − 1 1 + h [ (g2 + (h− 1)2)x2 1 − 2gx1y1 − hy21 ] . So we obtain a subfamily of the family of systems (1.3). 2: B2 = 0. Then we obtain α = β = 0 and considering (4.13) this condition yields a = 1[ g2 + (h− 1)2 ]2 [(e+ fg − eh)(2egh+ fh2 − f − fg2) ] ≡ a∗, b =− 1[ g2 + (h− 1)2 ]2 [efg(h− 1)(1 + 3h)− efg3 + (h− 1)2(f2 − e2h) + g2(f2 + e2h− 2f2h) ] ≡ b∗. In this case clearly we obtain systems (4.12) with a = a∗ and b = b∗ which we denote by (4.12∗). For these systems calculations yield B1 = B2 = 0 and B3 = 3[ g2 + (h− 1)2 ]2 (1+h)2(e+fg−eh)(f+fg2−2egh−fh2)(x2+y2)2. (4.14) We detect that systems (4.12∗) possess two complex invariant lines:[ g ± i(1− h) ] x+ (1− h∓ ig)y − (f ± ie) = 0 in two different directions (intersecting infinite line at complex singularities). Since for systems (1.3) we have θ = 8d1(b 2 1c2 − 4a0c 2 2 + 4a0c2d1 − a0d 2 1) ̸= 0, we deduce that in order to exist an affine transformation for bringing systems (4.12∗) to the form (1.3) we need a real invariant affine line in the third (real) direction. On the other hand according to Lemma 3.4 for the existence of invariant affine lines in three distinct directions, the condition B3 = 0 is necessary. So we conclude that in the case η < 0, θ ̸= 0, B1 = B2 = 0 and B3 ̸= 0 a quadratic system could not be brought to an Abel quadratic differential system. Assume now B3 = 0. Considering the condition θ ̸= 0 and (4.14) we obtain the condition (e+ fg − eh)(f + fg2 − 2egh− fh2) = 0. 2.1: e(1 − h) + fg = 0. If g = 0 then from θ ̸= 0 (i.e. g2 + (h − 1)2 ̸= 0) we obtain e = 0 and in this case systems (4.12∗) have the form ẋ = (h+ 1)xy, ẏ = − f2 (h− 1)2 + fy − x2 + hy2. Thus we obtain Abel quadratic systems of the form (1.3). Assume now g ̸= 0. Then we obtain f = e(h− 1)/g and systems (4.12∗) become ẋ = gx2 + (1 + h)xy, ẏ = −e2/g2 + e(h− 1)y/g − x2 + gxy + hy2. EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 15 Applying the transformation x1 = x, y1 = gx+ (1 + h)y we arrive at the following subfamily of the family of systems (1.3): ẋ1 =x1y1, ẏ1 =− e2(1 + h) g2 + 2ex1 + e(h− 1) g y1 + 1 1 + h [ (g2 + (h+ 1)2)x2 1 + 2gx1y1 + hy21 ] . 2.2: f(1 + g2 − h2)− 2egh = 0. If g = 0 then h2 − 1 ̸= 0 and we again get f = 0 and we arrive at the case considered above. If h = 0 then the condition f(1 + g2) = 0 gives f = 0 and this leads to the degenerate systems ẋ = x(gx+ y), ẏ = x(e− x+ gy). Assume now gh ̸= 0. Then we calculate e = f(1 + g2 − h2)/(2gh), and after the same transformation applied to systems (4.12∗) we obtain the systems ẋ1 =x1y1, ẏ1 = f2(1 + h) [ g2 + (h+ 1)2 ] 4g2h + −f(h− 1)(g2 + (h+ 1)2) 2gh x1 + fy1 − (g2 + (h+ 1)2) 1 + h x2 1 + 2g 1 + h x1y1 + h 1 + h y21 . Thus we obtain the Abel quadratic systems of the form (1.3). 4.2.2. Case θ = 0. According to (4.11) we have (h+ 1)[(h− 1)2 + g2] = 0, and we consider two subcases: Ñ ̸= 0 and Ñ = 0. 1: Ñ ̸= 0. Then by (4.11) the condition θ = 0 yields h = −1, and in addition we may assume f = 0 due to the translation (x, y) → (x, y + f/2). Hence we obtain the family of systems ẋ = a+ cx+ dy + gx2, ẏ = b+ ex− x2 + gxy − y2, (4.15) for which calculations yield: B1 = −d2g(α̂2 + β̂2), H7 = 4d(4 + g2), B2 = −648 [ α̂(α̂+ d2g) + β̂2 ] x4 + 648a(1 + h)2α̂y2(6x2 − y2) − 2592a(1 + h)2β̂xy(x2 − y2), (4.16) where α̂ =a(g2 − 4) + 4bg − 2ce− d(d+ e)g, β̂ =− 4ag + b(g2 − 4) + c2 + d2 − e2 + cdg. (4.17) We observe that the condition α̂ = β̂ = 0 is equivalent to B1 = B2 = 0, and so we examine two possibilities: B2 ̸= 0 and B2 = 0. 1.1: B2 ̸= 0. Then the condition B1 = 0 implies dg = 0, and we discuss two cases: H7 ̸= 0 and H7 = 0. 1.1.1: H7 ̸= 0. Considering (4.16) we have d ̸= 0, and this implies g = 0. So we obtain the family of systems ẋ = a+ cx+ dy, ẏ = b+ ex− x2 − y2, 16 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 and applying the transformation x1 = x, y1 = cx + dy + a and t1 = t/d we arrive at the family of systems ẋ1 = dy1, ẏ1 = bd2− a2+(d2e− 2ac)x1+(2a+ cd)y1− (c2+ d2)x2 1+2cx1y1− y21 . So we obtain a subfamily of the family of systems (1.3). 1.1.2: H7 = 0. Then d = 0 and we arrive at the systems ẋ = a+ cx+ gx2, ẏ = b+ ex− x2 + gxy − y2, (4.18) which could have real straight lines only in the direction x = 0. However the right hand side of the first equation does not have as a factor y. Comparing with systems (1.3) we deduce that there could not exist an affine transformation which brought the above systems to the form (1.3). 1.2: B2 = 0. Then we obtain α̂ = β̂ = 0 and considering (4.17) this condition yields a = 1 (4 + g2)2 (2c+ dg + eg)(−4e+ 2cg + dg2) ≡ a∗, b = 1 (4 + g2)2 [ cdg3 + (c2 − 3d2 − 4de− e2)g2 − 4c(d+ 2e)g − 4(c2 + d2 − e2) ] ≡ b∗. In this case clearly we obtain systems (4.15) with a = a∗ and b = b∗ which we denote by (4.15∗). For these systems calculations yield B1 = B2 = 0 and B3 = −3d2g(x2 + y2)2, H7 = 4d(4 + g2). (4.19) We detect that systems (4.15)∗ possess two complex invariant lines: (g ± 2i)x+ (2∓ ig)y + c∓ i(d+ e) = 0. We consider two cases: B3 ̸= 0 and B3 = 0. 1.2.1: B3 ̸= 0. In this case by the same arguments as earlier we deduce that in order to exist an affine transformation for bringing systems (4.15∗) to the form (1.3) we need a real invariant affine line in the third (real) direction. However according to Lemma 3.4 for the existence of invariant affine lines in three distinct directions the condition B3 = 0 is necessary. So we conclude that in the case considered, a quadratic system (4.15)∗ could not be brought to an Abel quadratic system of the form (1.3). 1.2.2: B3 = 0. Considering (4.19) the condition dg = 0 holds, and we examine two subcases: H7 ̸= 0 and H7 = 0. 1.2.2.1: H7 ̸= 0. Then d ̸= 0 and the condition B3 = 0 implies g = 0. Then systems (4.15∗) become ẋ = −ce/2 + cx+ dy, ẏ = (c2 + d2 − e2)/4 + ex− x2 − y2, and applying the transformation x1 = x, y1 = cx+dy−ce/2 and t1 = t/d we arrive at the following subfamily of systems (1.3): ẋ1 =dy1, ẏ1 =(c2 + d2)(d2 − e2)/4 + (c2 + d2)ex1 + c(d− e)y1 − (c2 + d2)x2 1 + 2cx1y1 − y21 . EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 17 1.2.2.2: H7 = 0. Then d = 0 which implies B3 = 0. In this case considering systems (4.15∗) we arrive at the systems ẋ = a∗ ∣∣ d=0 + cx+ gx2, ẏ = b∗ ∣∣ d=0 + ex− x2 + gxy − y2. These systems could possess invariant lines in the unique real direction x = 0. However by the same arguments as we present earlier for systems (4.18) we conclude that there could not exist an affine transformation which brings the above systems to the form (1.3). 2: Ñ = 0. Then from (4.11) we have g = h− 1 = 0 and without loss of generality we may assume c = d = 0 via the translation (x, y) → (x− d/2, y− c/2). Hence we obtain the systems ẋ = a+ 2xy, ẏ = b+ ex+ fy − x2 + y2, (4.20) for which calculations yield: B1 =− 4a(e2 + f2)2, B2 =− 648 [ (e4 − 8aef + 2e2f2 + f4)x4 + 16a(e2 − f2)x3y + 48aefx2y2 + 16a(f2 − e2)xy3 − 8aefy4 ] . We observe that the condition e = f = 0 is equivalent to B1 = B2 = 0, and so we consider two possibilities: B2 ̸= 0 and B2 = 0. 2.1: B2 ̸= 0. In this case the condition B1 = 0 implies a = 0 and evidently systems (4.20) are of the form (1.3). 2.2: B2 = 0. Then considering the condition B1 = 0 we obtain e = f = 0, and we obtain the family of systems ẋ = a+ 2xy, ẏ = b− x2 + y2, for which we have B3 = −12a(x2 + y2)2. We detect that these systems possess the following two couples of complex invariant lines: b+ ia− (x− iy)2 = 0, b− ia− (x+ iy)2 = 0. According to Lemma 3.4 if B3 ̸= 0 then in the real direction x = 0 the above systems do not have any invariant line and this means that we could not bring them to the form (1.3) via an affine transformation. It remains to observe that for B3 = 0 (i.e. a = 0) the above systems are of the form (1.3). Thus all the possibilities in the case η < 0 are examined and we conclude that the statement (B) of Theorem 1.2 is proved. 4.3. The subfamily defined by (C): η = 0, M̃ ̸= 0. In this case by Lemma 3.2 we have to consider the systems (SIII) for which calculations yield: θ = 8h2(1− g), µ0 = gh2, Ñ = (g2 − 1)x2 + 2h(g − 1)xy + h2y2. (4.21) We consider two cases: θ ̸= 0 and θ = 0. 18 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 4.3.1. Case θ ̸= 0. Then (g−1)h ̸= 0 and by a translation we can assume d = e = 0, i.e. we obtain the systems ẋ = a+ cx+ gx2 + hxy, ẏ = b+ fy + (g − 1)xy + hy2, (4.22) for which we calculate B1 = −a2b(g − 1)2h4. Therefore due to θ ̸= 0 the condition B1 = 0 implies ab = 0 and we consider two subcases: a = 0 or b = 0. 1: a = 0. In this case applying the transformation x1 = x, y1 = gx + hy + c we arrive at the family of systems ẋ1 = x1y1, ẏ1 = c2 − cf + bh+ (c+ cg − fg)x1 + (f − 2c)y1 + gx2 1 − x1y1 + y21 , which is a subfamily of (1.3). 2: b = 0. Then systems (4.22) possess the invariant line y = 0 and using the transformation x1 = y, y1 = (g − 1)x+ hy + f we arrive at the systems ẋ1 = x1y1, ẏ1 = b′ + e′x1 + l′x2 1 + (f ′ + 2m′x1)y1 + n′y21 , where b′, e′, f ′, l′,m′ and n′ are rational functions of the parameters a, c, f, g, h with the same denominator g−1 ̸= 0. So we obtained the systems belonging to the family of systems (1.3). 4.3.2. Case θ = 0. By (4.21) we obtain h(g−1) = 0, and since µ0 = gh2 we consider two subcases: µ0 ̸= 0 and µ0 = 0. 1: µ0 ̸= 0. Considering (4.21) we obtain h ̸= 0, g = 1, and then we may assume h = 1 due to the rescaling y → y/h. Moreover we may assume c = d = 0 via the translation (x, y) → (x− d, y + 2d− c). So, we obtain the systems ẋ = a+ x2 + xy, ẏ = b+ ex+ fy + y2, for which calculation yields B1 = −a2e2, B2 = 648 [ (4a− b)e2x4 + 4ae2x3y − a2y4 ] , H7 = −4e. The condition B1 = 0 implies ae = 0, and we consider two possibilities: H7 ̸= 0 and H7 = 0. 1.1: H7 ̸= 0. In this case e ̸= 0 and we obtain a = 0. Then applying the transformation x1 = x, y1 = x+ y we arrive at the family of systems ẋ1 = x1y1, ẏ1 = b+ (e− f)x1 + fy1 + x2 1 − x1y1 + y21 , (4.23) which is a subfamily of (1.3). 1.2: H7 = 0. Then e = 0 and this leads to the systems ẋ = a+ x2 + xy, ẏ = b+ fy + y2, for which B1 = 0, B2 = −648a2y4. We observe that these systems possess only two (parallel) invariant lines b+ fy + y2 = 0 in the direction y = 0 which could be real or complex or could coincide. Moreover by Lemma 3.4 in the case B2 ̸= 0 we do not have any other invariant line in the second direction x = 0. Therefore by the same arguments as we presented earlier for systems (4.18), we conclude that for B2 ̸= 0 there cannot exist an affine transformation which brings the above systems to the form (1.3). EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 19 Assuming B2 = 0 we obtain a = 0 and using the transformation x1 = x, y1 = x+ y we arrive at the systems (4.23) with e = 0, i.e. we obtain systems of the form (1.3). 2: µ0 = 0. Since θ = 0 this implies h = 0 and for the systems (SIII) we have Ñ = (g2 − 1)x2 and we examine two possibilities: Ñ ̸= 0 and Ñ = 0. 2.1: Ñ ̸= 0. In this case g − 1 ̸= 0, and we may assume e = f = 0 via the translation (x, y) → ( x+ f/(1− g), y + e/(1− g) ) . This leads to the systems ẋ = a+ cx+ dy + gx2, ẏ = b+ (g − 1)xy, (4.24) for which we have B1 = −bd4(g − 1)2g2, Ñ = (g2 − 1)x2, H7 = 4d(g2 − 1). Because Ñ ̸= 0 the condition B1 = 0 gives bdg = 0 and we consider two cases: H7 ̸= 0 and H7 = 0. 2.1.1: H7 ̸= 0. Then d ̸= 0 and we obtain bg = 0. If b = 0 then it is evident that after the interchange x ↔ y systems (4.24) become of the form (1.3). Assume now g = 0. Since d ̸= 0 we can apply the transformation x1 = x, y1 = cx+ dy + a and this leads to the following subfamily of (1.3): ẋ1 = y1, ẏ1 = bd+ ax1 + cy1 + cx2 1 − x1y1. 2.1.2: H7 = 0. Then d = 0 and we obtain the systems ẋ = a+ cx+ gx2, ẏ = b+ (g − 1)xy, (4.25) which possess the invariant lines a+ cx+gx2 = 0 in the real double direction x = 0 because C2 = x2y. We calculate B1 = B2 = 0, B3 = −3b(g − 1)2x4, and we conclude that an invariant line exists in the direction y = 0 if and only if B3 = 0. So by the same arguments as we presented earlier for systems (4.18), we conclude that for B3 ̸= 0 there cannot exist an affine transformation which brings the above systems to the form (1.3). Assuming B3 = 0 we obtain b = 0 (due to Ñ ̸= 0) and in the same manner as above by the interchange x ↔ y systems (4.24) become of the form (1.3). 2.2: Ñ = 0. In this case g2 − 1 = 0, i.e. g = 1 or g = −1. On the other hand for systems (SIII) with h = 0 we have K̃ = g(g − 1)x2 and we consider two cases: K̃ ̸= 0 and K̃ = 0. 2.2.1: K̃ ̸= 0. Then g − 1 ̸= 0 and this implies g = −1. In this case we may assume e = f = 0 via the translation (x, y) → ( x+ f/2, y + e/2 ) and we arrive at the family of systems ẋ = a+ cx+ dy − x2, ẏ = b− 2xy, (4.26) for which calculations yield B1 = −4bd4, θ3 = 2d2. 20 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 2.2.1.1: θ3 ̸= 0. Then the condition B1 = 0 gives b = 0 and after the interchange x ↔ y the above systems become of the form (1.3). 2.2.1.1: θ3 = 0. This implies d = 0 and we obtain the systems (4.25) with g = −1. So we repeat the same steps as before in this particular case and we conclude that the systems (4.26) could be brought via an affine transformation to the form (1.3) if and only if either θ3 ̸= 0, or θ3 = 0 and B3 = 0. 2.2.2: K̃ = 0. Then g = 1 and we may assume c = 0 due to the translation (x, y) → (x− c/2, y). Then we obtain the systems ẋ = a+ dy + x2, ẏ = b+ ex+ fy. It is clear that in order to have invariant lines in the direction x = 0 (respectively y = 0) the condition d = 0 (respectively e = 0) has to be satisfied. Assume first d = 0. Then we obtain two parallel invariant lines and clearly we could not use them for the construction of the transformation which brings these systems to the form (1.3). So we need a real invariant line in the direction y = 0, i.e. the condition e = 0 must hold. In this case we obtain the invariant line fy + b = 0 if f ̸= 0. However applying the transformation x1 = fy + b, y1 = γx+ δy + ν with free parameters γ, δ and ν, we arrive at the systems ẋ1 = fx1, ẏ1 = Q̃(x1, y1). As it can be observed these systems do not have the form (1.3). So we deduce that in the case Ñ = 0 = K̃ there cannot exist an affine trans- formation which brings a system (SIII) to an Abel quadratic system of the form (1.3). Since all the possibilities in the case η = 0 and M̃ ̸= 0 are examined we have that the statement (C) of Theorem 1.2 is proved. 4.4. Subfamily defined by (D): η = M̃ = 0. According to the conditions pro- vided by Theorem 1.2 we consider two cases: C2 ̸= 0 and C2 = 0. 4.4.1. Case C2 ̸= 0. Then by Lemma 3.2 we examine the systems (SIV ) for which calculations yield: η = M̃ = 0, C2 = x3, θ = 8h3. We consider two subcases: θ ̸= 0 and θ = 0. 1: θ ̸= 0. Then h ̸= 0 and by a translation we can assume c = d = 0, i.e. we obtain the systems ẋ = a+ gx2 + hxy, ẏ = b+ ex+ fy − x2 + gxy + hy2, (4.27) for which we calculate B1 = −a3h6. So the condition B1 = 0 gives a = 0 and then the above systems after the transformation x1 = x, y1 = gx+ hy become ẋ1 = x1y1, ẏ1 = bh+ (eh− fg)x1 + fy1 − hx2 1 + y21 , (4.28) i.e. we obtain a subfamily of (1.3). 2: θ = 0. Then h = 0 and we calculate B1 = −d6g3, Ñ = g2x2 and we consider two possibilities: Ñ ̸= 0 and Ñ = 0. EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 21 2.1: Ñ ̸= 0. We have g ̸= 0 and the condition B1 = 0 gives d = 0. In this case due to a translation we may assume e = f = 0, and this leads to the systems ẋ = a+ cx+ gx2, ẏ = b− x2 + gxy. (4.29) Since for these systems we have C2 = x3 (i.e. we could have real invariant affine lines only in this direction) we conclude, that besides the parallel invariant lines a+ cx+ gx2 = 0 the above systems cannot have other invariant lines. Thus applying the same arguments as we present earlier for systems (4.18), we deduce that for Ñ ̸= 0 there cannot exist an affine transformation which brings systems (4.29) to the form (1.3). 2.2: Ñ = 0. Then g = 0 (this implies B1 = 0) and we arrive at the systems ẋ = a+ cx+ dy, ẏ = b+ ex+ fy − x2, for which B2 = −648d4x4. 2.2.1: B2 ̸= 0. We obtain d ̸= 0 and applying the transformation x1 = x, y1 = cx+ dy + a we obtain the following subfamily of (1.3): ẋ1 = y1, ẏ1 = bd− af + (de− cf)x1 + (c+ f)y1 − dx2 1. (4.30) 2.2.2: B2 = 0. Then we obtain the systems ẋ = a+ cx, ẏ = b+ ex+ fy − x2, and since the right hand side of the first equation does not have as a factor y we deduce that there could not exist an affine transformation which brings the above systems to the form (1.3). 4.4.2. Case C2 = 0. Then by Lemma 3.2 we examine the systems (SV ) which have the infinite line filled up with singularities. This family of systems is considered in [18], where a total of 9 canonical forms are given for this family: C2.1 – C2.9 (see Table 1, page 741). We observe that the canonical systems C2.1 – C2.4 for H10 ̸= 0 as well as C2.5 – C2.7 for H10 = 0 and H12 ̸= 0 after the additional interchange x ↔ y have the form ẋ = xy, ẏ = Qi(x, y), (i = 1, . . . , 7) where Qi(x, y) is the corresponding to C2.i quadratic polynomial depending of at least one parameter. It is evident that these canonical systems belong to the family (1.3). It remains to consider two canonical systems given in Table 1 of [18]: (C2.8) : { ẋ = x+ x2, ẏ = 1 + xy; (C2.9) : { ẋ = x2, ẏ = 1 + xy, and we claim that there does not exist an affine transformation bringing any of these two systems to the form (1.3). Indeed, for both systems (C2.8) and (C2.9) we have: C2 = 0 and H10 = 0 = H12. On the other hand for systems (1.3) we calculate C2 = −c2x 3 − b1x 2y + (d1 − a0)xy 2, 22 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 and hence the condition C2 = 0 implies c2 = b1 = 0 and d1 = a0. Then we obtain the systems ẋ = (d0 + a0x)y, ẏ = c0 + c1x+ b0y + a0x 2, for which calculations yield H10 = 36a40c 2 1 = 0, H12 ∣∣ {a0c1=0} = −8a40c 2 0y 2 = 0. If a0 ̸= 0 then we obtain c1 = 0 = c0 and this leads to the degenerate systems ẋ = (d0 + a0x)y, ẏ = y(b0 + a0y). On the other hand assuming a0 = 0 we obtain the linear systems ẋ = d0y, ẏ = c0 + c1x+ b0y. This completes the proof of our claim. Thus all the cases are examined and the proof of Theorem 1.2 is complete. □ 4.5. Phase portraits of the quadratic systems from the family (D) defined by Theorem 1.2. According to Lemma 3.2 the systems from the family (D) de- fined by the condition η = M̃ = 0 could be brought via an affine transformation either to the systems (SIV ) (if C2 ̸= 0), or to the systems (SV ) (if C2 = 0). So we examine these two subfamilies separately. We give examples for the realization of each one of the phase portraits of systems (1.1) constructed, and which belong to one of the above mentioned two classes in the form (a, c, d, g, h, k), (b, e, f, l,m, n). 4.5.1. Systems (SIV ): η = M̃ = 0, C2 ̸= 0. Theorem 4.1. Assume that for a quadratic system the conditions η = M̃ = 0, and C2 ̸= 0 hold. In agreement to Theorem 1.2 this system belongs to the class QSAb if and only if either θ ̸= 0, or θ = Ñ = 0 and B2 ̸= 0. In this case its phase portrait is topologically equivalent to one of the pictures given in Figure 1 if and only if the following corresponding conditions are verified: Picture SIV .1 ⇔ θ ̸= 0, D < 0, R > 0, S > 0, µ0 < 0, ¬(C2), Ũ1Ũ2 < 0; (3) Picture SIV .2 ⇔ θ ̸= 0, D < 0, R > 0, S > 0, µ0 < 0, ¬(C2), Ũ1Ũ2 > 0; (3) Picture SIV .3 ⇔ θ ̸= 0, D < 0, R > 0, S > 0, µ0 < 0, (C2); (4) Picture SIV .4 ⇔ θ ̸= 0, D < 0, R > 0, S > 0, µ0 > 0; (8) Picture SIV .5 ⇔ { θ ̸= 0,D < 0, (R ≤ 0) ∨ (S ≤ 0) or θ = Ñ = 0, B2 ̸= 0,U < 0; (12) Picture SIV .6 ⇔ { θ ̸= 0,D > 0, µ0 < 0,¬(Ĉ1), Ũ1 < 0, or θ ̸= 0,D = T = P = 0,R ̸= 0, T4 ̸= 0, µ0 < 0; (16) Picture SIV .7 ⇔ θ ̸= 0, D > 0, µ0 < 0, ¬(Ĉ1), Ũ1 > 0; (16) Picture SIV .8 ⇔ θ ̸= 0, D > 0, µ0 < 0, (Ĉ1); (18) Picture SIV .9 ⇔ { θ ̸= 0,D > 0, µ0 > 0,¬(C2), or θ ̸= 0,D = T = P = 0,R ̸= 0, T4 ̸= 0, µ0 > 0, or θ = Ñ = 0, B2 ̸= 0,U > 0,¬(C9); (23) Picture SIV .10 ⇔ { θ ̸= 0,D > 0, µ0 > 0, (C2), or θ = Ñ = 0, B2 ̸= 0,U > 0, (C9); (24) Picture SIV .11 ⇔ θ ̸= 0, D = 0, T < 0 µ0 < 0, B2Ũ1 ̸= 0, E1 ̸= 0; (30) Picture SIV .12 ⇔ θ ̸= 0, D = 0, T < 0 µ0 < 0, B2Ũ1 ̸= 0, E1 = 0; (34) Picture SIV .13 ⇔ θ ̸= 0, D = 0, T < 0 µ0 < 0, B2 = 0; (30) Picture SIV .14 ⇔ θ ̸= 0, D = 0, T < 0 µ0 < 0, Ũ1 = 0; (30) EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 23 Picture SIV .15 ⇔ θ ̸= 0, D = 0, T < 0 µ0 > 0; (37) Picture SIV .16 ⇔ { θ ̸= 0,D = 0,T > 0, E1 ̸= 0 or θ ̸= 0,D = T = P = R = 0, µ0 > 0; (44) Picture SIV .17 ⇔ { θ ̸= 0,D = 0,T > 0, E1 = 0 or θ = Ñ = 0, B2 ̸= 0,U = 0; (47) Picture SIV .18 ⇔ θ ̸= 0, D = 0, T = 0 P ̸= 0; (50) Picture SIV .19 ⇔ θ ̸= 0, D = 0, T = 0 P = 0, R ̸= 0, T4 = 0, µ0 < 0; (60) Picture SIV .20 ⇔ θ ̸= 0, D = 0, T = 0 P = 0, R ̸= 0, T4 = 0, µ0 > 0; (64) Picture SIV .21 ⇔ θ ̸= 0, D = 0, T = 0 P = 0, R = 0, µ0 < 0. (67) The right-most entry in the table above corresponds to the global topological con- figurations of the singularities (finite and infinite), according to the notation in the set of diagrams provided by [4, Main Theorem]. Figure 1. Global phase portraits of quadratic systems with η = M̃ = 0, C2 ̸= 0. 24 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 Proof. We prove this theorem following the conditions provided by the statement (D) of Theorem 1.2 in the case C2 ̸= 0. So we discuss two cases: θ ̸= 0 and θ = Ñ = 0, B2 ̸= 0. 1: Case θ ̸= 0. Considering the systems (4.27) and the corresponding transformed systems (4.28) we examine the family: ẋ = xy ≡ P (x, y), ẏ = b+ ex+ fy − hx2 + y2 ≡ Q(x, y). (4.31) We shall consider step by step the conditions provided by the Diagrams 1-6 from [4], taking into account that the conditions η = M̃ = 0, and C2 ̸= 0 are satisfied. For these systems calculations yield: C2 = hx3, µ0 = −h, D = −48b2(f2 − 4b)(e2 + 4bh), B2 = −648b2h2x4, T4 = −f2h(9b− 2f2), T3 = −fh(18b− 5f2), T2 = −3(3b− f2)h, F = 9fh/8, F1 = 0, F2 = −9f2h2/2 = −F3, B = −9(9e2 + 36bh− 8f2h)/8, H = −9h/2, σ = f + 3y. (4.32) We observe that condition C2 ̸= 0 implies µ0 ̸= 0, and according with [5, Table 6.2] the above systems possess finite singularities of total multiplicity four. More exactly we have the singularities M1,2(0, y1,2) and M3,4(x3,4, 0), where y1,2 = ( − f ± √ f2 − 4b ) /2, x3,4 = ( e± √ e2 + 4bh ) /(2h). (4.33) It is clear that the singularities M1,2(0, y1,2) are located on the real invariant line x = 0, and these singularities are real if f2 − 4b > 0 and they are complex if f2 − 4b < 0. First we prove the following lemma: Lemma 4.2. For a system (4.31) the conditions (C1) as well as the conditions (C5) - (C10) and (Ĉ2) could not be satisfied. Proof. First, from (4.32) we obtain that for systems (4.31) the condition σ = f + 3y ̸= 0 holds. Therefore considering (3.1) and (3.2) we deduce that the conditions (C5) - (C10) and (Ĉ2) could not be satisfied for these systems. It remains to examine the conditions (C1). According to (3.1) these conditions imply T3 ̸= 0 and F2 = 0. However considering (4.32) it is clear that the condition T3 ̸= 0 (i.e. fh ̸= 0) implies F2 ̸= 0. This completes the proof of the lemma. □ According to [5, Table 6.2] all the finite singularities of systems (4.31) are distinct if D ̸= 0 and we have multiple singular points if D = 0. So we examine three subcases: D < 0, D > 0 and D = 0. 1.1: D < 0. According to [5, Table 6.2] systems (4.31) possess either four real distinct finite singularities in the case R > 0 and S > 0, or four complex finite singularities if (R ≤ 0) ∨ (S ≤ 0). 1.1.1: R > 0 and S > 0. So systems (4.31) possess four real distinct finite singularities and following [4, Diagram 1, page 3] we consider two cases: µ0 < 0 and µ0 > 0. 1.1.1.1: µ0 < 0. According to this diagram we could either have the topo- logical configuration (3) s, a, a, a;S if ¬(C2), or (4) s, a, a, c;S if (C2). EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 25 Consider first the configuration (3). It is clear that if the saddle is located on the invariant line x = 0 then we have the separatrix connection between the finite saddle and the infinite one. So we need a condition to distinguish whether the saddle is located on the invariant line or not. On the other hand denoting by ∆i (i = 1, 2, 3, 4) the determinant of the linear matrix corresponding to the singular point Mi we calculate ∆1,2 = −2b+ (f2 ± f √ f2 − 4b )/2 ⇒ ∆1∆2 = b(4b− f2). We remark that when two finite singularities coalesce (M1 with M2 if 4b−f2 = 0, or M3 with M4 if e2 +4bh = 0, or M1 with M4 if b = 0) it is important to distinguish if both singularities are located on the invariant line. For systems (4.31) we have: Ũ1 = −27(f2 − 4b)h/8, Ũ2 = 9bh/2. (4.34) Therefore Ũ1Ũ2 = 243b(4b− f2)h2/16 = 243∆1∆2h 2/16 and we conclude that the following remark is valid: Remark 4.3. Assume that the singularities of a system (4.31) located on the invariant line x = 0 are real and in addition the condition Ũ1Ũ2 ̸= 0 holds. Then sign(∆1∆2) = sign(Ũ1Ũ2), i.e. on the invariant line of this system lies exactly one saddle if and only if Ũ1Ũ2 < 0. So, considering the above remark and the fact that we have a single saddle, in the case of the topological configuration (3) s, a, a, a;S (see [4]) we obtain Picture SIV .1 if Ũ1Ũ2 < 0 and Picture SIV .2 if Ũ1Ũ2 > 0. The corresponding examples are: Picture SIV .1 if ¬(C2) and Ũ1Ũ2 < 0 [Ex: (0, 0, 0, 0, 1/2, 0), (1/8, 0, 1,−1, 0, 1)]; Picture SIV .2 if ¬(C2) and Ũ1Ũ2 > 0 [Ex: (0, 0, 0, 0, 1/2, 0), (−3/4, 2, 1,−1, 0, 1)]; Consider now the configuration (4): s, a, a, c;S. Since we have a center (i.e. the conditions (C2) hold), considering [20] (see also [21]) we obtain the unique phase portrait given by Picture SIV .3 [Ex: (0, 0, 0, 0, 1/2, 0), (−1, √ 5, 0,−1, 0, 1)]. 1.1.1.2: µ0 > 0. In this case from [4, Diagram 1, page 3] we could either have the topological configuration (8) s, s, a, a;N if ¬((Ĉ1) ∨ (Ĉ2)), or configuration (9) s, s, c, c;N if (Ĉ1) ∨ (Ĉ2). We claim that configuration (9) with two centers is not realizable for systems (4.31). According to Lemma 4.2 the conditions (Ĉ2) are ruled out for these systems. Consider now the conditions (Ĉ1). According to (4.32) and (3.2) the condition H = −9h/2 < 0 is necessary, but this implies h > 0 which contradicts µ0 = −h > 0. This completes the proof of the claim. It remains to examine the configuration (8) s, s, a, a;N . It is not too difficult to convince ourself that both saddles could not be located on the invariant line x = 0 (since ∆1 +∆2 = f2 − 4b > 0 by (4.33)). If both singularities on x = 0 are nodes, then we obtain Picture SIV .4 [Ex: (0, 0, 0, 0, 1/2, 0), (−3/4, 1, 1, 1, 0, 1)]. Assume now that on the invariant line x = 0 we have a saddle and a node. Lemma 4.4. Systems (4.31) with configuration s, s, a, a;N and both a saddle and a node on x = 0 possess only one phase portrait which is topologically equivalent to Picture SIV .4. Proof. First of all we simplify the canonical systems (4.31) using the corresponding conditions for this case. Since we assume that systems (4.31) with µ0 > 0 have four 26 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 real singularities and in addition on the invariant line x = 0 we have a node and a saddle then, as it was mentioned earlier, for these systems the conditions h < 0, f2 − 4b > 0, e2 + 4bh > 0, ∆1∆2 = b(4b− f2) < 0, hold. These conditions imply b > 0 and ef ̸= 0, and therefore we may assume h = −1 = e due to the rescaling (x, y, t) 7→ (ex/h, −ey/ √ −h, − √ −h t/e). So we arrive at the 2-parameter family of systems ẋ = xy, ẏ = b− x+ fy + x2 + y2 (4.35) for which in addition we may assume f < 0 due to the rescaling (x, y, t) 7→ (x,−y,−t). Therefore considering the above conditions depending on four param- eters, for systems (4.35) the conditions 0 < b < 1/4 and f < −2 √ b are satisfied. We recall that the above systems possess four real singularities M1,2(0, y1,2) and M3,4(x3,4, 0) with the coordinates y1,2 = ( − f ± √ f2 − 4b ) /2, x3,4 = ( 1∓ √ 1− 4b ) /2). Since 0 < b < 1/4, f < −2 √ b, ∆1∆2 < 0 and ∆3∆4 < 0, it is not too difficult to show that the following conditions hold: 0 < y2 < y1, ∆2 < 0 < ∆1, 0 < x3 < x4, ∆4 < 0 < ∆3. In other words the two saddles are located at M2 and M4 which is in accordance with Berlinskĭi theorem [8]. We observe that the isocline on which ẏ = 0 is a real ellipse, because the de- terminant of the conic is (−1 + 4b − f2)/4 < 0 due to the condition f2 − 4b > 0. We plot this isocline in dotted line as well as the isocline ẋ = xy = 0 and the line conecting the two saddles M2 and M4 (see Figure 2). We claim that the non vertical eigenvector of the saddle M2 has a negative slope SlM2 which is smaller that the slope SlL of the line joining both saddles, and it is bigger than the slope SlTE of the tangent to the ellipse at the point M2. Indeed, it is not difficult to determine that SlM2 = √ 1− 4b+ 1 f − √ f2 − 4b , SlL = 2 f − √ f2 − 4b , SlTE = − 1√ f2 − 4b , and then we obtain SlM2−SlL = √ 1− 4b+ 1 f − √ f2 − 4b < 0, SlM2−SlTE = − √ f2 − 4b+ f√ f2 − 4b (√ f2 − 4b− f ) > 0, because of the conditions 0 < b < 1/4 and f < −2 √ b. This completes the proof of the claim. On the other hand we observe that the line connecting the two saddles, i.e. the line y = SlLx+m, m = 2b√ f2 − 4b− f , is an isocline for systems (4.35). Indeed, calculations yield dy dx ∣∣ y=SlLx+m = b− x+ x2 + fy + y2 xy ∣∣ y=SlLx+m = f − √ 1− 4b √ f2 − 4b 2b . EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 27 Moreover we detect that the slope of the line is greater than the slope of the flow on the line, because we have SlL − f − √ 1− 4b √ f2 − 4b 2b = ( √ 1− 4b+ 1)( √ f2 − 4b− f) 4b > 0. We remark that all this information is presented in Figure 2. M M M M 1 2 3 4 Figure 2. A scheme of the flow on the phase plane of systems (4.35). Thus the separatrix of the saddle M2 located in the first quadrant cannot go to the infinite singularity [0 : 1 : 0], and hence it can only either go to the anti-saddle M3, or connect with a separatrix of M4. In the first case we arrive in the unique way to the phase portrait equivalent to Picture SIV .4. We claim that the connection of M2 with M4 is not possible. Indeed, we observe that the ellipse and the isocline y = 0 intersect at the saddle M4 producing four regions around M4. It is easy to determine that due to the horizontal flow on the ellipse and the vertical flow on the isocline y = 0, in each one of these regions there must be one separatrix of this saddle in each of the four regions. But the separatrix which is inside the ellipse and over the x-axis (which is the separatrix supposed to produce the connection) cannot be below the line joining the saddles because of the flow along all these lines. This proves the claim and completes the proof of the lemma. □ 1.1.2: (R ≤ 0) ∨ (S ≤ 0). It was mentioned earlier that in this case we have four complex singularities. According to [4, Diagram 1, page 3] we could 28 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 have a single topological configuration (12) N , which leads to Picture SIV .5: [Ex: (0, 0, 0, 0, 1/2, 0), (1, 0, 0, 1, 0, 1)]. 1.2: D > 0. According to [5, Table 6.2] systems (4.31) possess two real and two complex finite singularities. Considering the coordinates (4.33) of the singularities M1,2(0, y1,2) we observe that they are real if f2 − 4b > 0, and they are complex if f2 − 4b < 0. On the other hand for systems (4.31) we have Ũ1 = −27(f2−4b)h/8 and µ0 = −h and hence, sign(f2 − 4b) = sign(µ0Ũ1). So we arrive at the next remark. Remark 4.5. Assume that for a quadratic system (4.31) the condition Dµ0 ̸= 0 holds. Then the finite singularities located on the invariant line x = 0 of this system are real if µ0Ũ1 > 0, and they are complex if µ0Ũ1 < 0. 1.2.1: µ0 < 0. Since η = M̃ = 0, by [4, Diagram 1, page 4] systems (4.31) could either have the topological configuration (16) a, a; if ¬((C1) ∨ (Ĉ1)), or (17) a, c;S if (C1), or (18) c, c;S if (Ĉ1). We observe that by Lemma 4.2 the conditions (C1) from (3.1) are incompatible with systems (4.31). This means that the topological configuration (17) could not be realized for these systems. Since µ0 < 0, considering Remark 4.5 it is not difficult to show that in the case of the configuration (16) a, a;S we obtain Picture SIV .6 if Ũ1 < 0 and Picture SIV .7 if Ũ1 > 0. On the other hand the configuration (18) c, c;S leads to the Picture SIV .8. We exhibit three examples of realization of the pictures: Picture SIV .6: [Ex: (0, 0, 0, 0, 1/2, 0), (−1, 2, 1,−2, 0, 1)]; Picture SIV .7: [Ex: (0, 0, 0, 0, 1/2, 0), (2, 0,−1,−1, 0, 1)]; Picture SIV .8: [Ex: (0, 0, 0, 0, 1/2, 0), (1, 0, 0,−1, 0, 1)]. 1.2.2: µ0 > 0. Considering the condition η = M̃ = 0, by [4, Diagram 1, page 4] systems (4.31) could either have the topological configuration (23) s, a;N if ¬((C2)∨ (C7)), or (24) s, c;N (C2)∨ (C7). However by Lemma 4.2 the conditions (C7) could not be satisfied. Thus considering Remark 4.5 and the condition µ0 > 0 we deduce that the configuration (23) s, a;N with the condition ¬(C2) leads to Picture SIV .9 if Ũ1 > 0. If Ũ1 < 0 then by Remark 4.5 the real finite singularities are located outside the invariant straight line and hence the anti-saddle could be a focus. Therefore we conclude that there could be three topologically distinct phase portraits presented in Figure 3. In the case when the anti-saddle is a node only the phase portrait given by Picture SIV .9 (which is topologically equivalent to Picture (a) in Figure 3) is possible. But when besides the saddle we have a focus, then we could have two more topologically distinct possibilities, given by Picture (b) (possessing a loop) and Picture (c) (possessing a limit cycle). Lemma 4.6. The phase portraits given by Pictures (b) and (c) of Figure 3 cannot be realizable for systems (4.31). Proof. By Dulac’s theorem (see [9, Theorem 7.12]) since taking the functionB(x, y) = x−3 in the simply connected region R = {x > 0} (or R = {x < 0} ), the divergence ∂(BP )/∂x + ∂(BQ)/∂y = fx−3 is strictly positive or negative in R if f ̸= 0 with EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 29 P and Q given as in systems (4.31), these systems cannot have periodic orbits ly- ing entirely in R. This proves that Picture (c) cannot be realizable when f ̸= 0. Moreover since the proof of Dulac’s theorem also works for a homoclinic loop and its singular point, it follows that in the case f ̸= 0 Picture (b) is also non realizable. Assume now f = 0. Then the systems ẋ = BP and ẏ = BQ are Hamiltonian in R, because their divergence is identically zero. So these systems cannot have a focus, and consequently systems (4.31) also cannot have a focus. This completes the proof of the lemma. □ It is clear that the configuration (24) s, c;N (with the conditions (C2)) leads to Picture SIV .10. The realization of the phase portraits in the case µ0 > 0 is proved by the following examples: Picture SIV .9: [Ex: (0, 0, 0, 0, 1/2, 0), (1/8, 0, 1, 1, 0, 1)]; Picture (a), Figure 3: [Ex: (0, 0, 0, 0, 1/2, 0), (2, 3,−1, 1, 0, 1)] ∼= Picture SIV .9; Picture SIV .10: [Ex: (0, 0, 0, 0, 1/2, 0), (1/2, 2, 0, 1, 0, 1)]. Figure 3. Some phase portraits of quadratic systems with η = M̃ = 0, C2 ̸= 0. 1.3: D = 0. If T ̸= 0 then according to [5, Table 6.2] systems (4.31) possess one double real singular point and two distinct finite singularities. Moreover these two singular points are real if T < 0 and complex if T > 0. In the case T = 0 and µ0 ̸= 0 by [5, Table 6.2] these systems possess at most two finite singularities of total multiplicity four. Considering (4.32) we detect that the condition D = 0 gives three possibilities: (1) b = 0; (2) f2−4b = 0 and (3) e2+4bh = 0. Taking into account the values of the invariant polynomials D, B2 and Ũ1 from (4.32) and (4.34) it is easy to determine, that due to µ0 ̸= 0 the three mentioned possibilities could be distinguished by means of these invariant polynomials. More precisely, considering also the coordinates (4.33) of the finite singularities M1,2(0, y1,2) and M3,4(x3,4, 0) we have the next remark. Remark 4.7. (i) The following conditions are equivalent: (1) b = 0 ⇔ B2 = 0; (2) f2 − 4b = 0 ⇔ Ũ1 = 0; (3) e2 + 4bh = 0 ⇔ D = 0 and B2Ũ1 ̸= 0. 30 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 (ii) In the case B2 = 0 (respectively Ũ1 = 0; D = 0, B2Ũ1 ̸= 0) the singular point M4 coalesces with M1 (respectively M2 with M1; M4 with M3). (iii) the condition B2 = 0 (i.e. b = 0) implies T ≤ 0 since T = −3e2f2x2y2(fhx− ey)2. In what follows we consider three cases: T < 0, T > 0 and T = 0. 1.3.1: T < 0. Then all three finite singularities (one of them is double) are real and following [4, Diagram 1, page 4] we consider two cases: µ0 < 0 and µ0 > 0. 1.3.1.1: µ0 < 0. Since η = M̃ = 0, according to this diagram we could either have the topological configuration (30) a, a, sn;S if E1 ̸= 0, or (34) a, a, cp;S if E1 = 0. Considering Remark 4.7 (i) we examine three subcases: B2Ũ1 ̸= 0; B2 = 0 and Ũ1 = 0. 1.3.1.1.1: B2Ũ1 ̸= 0. Then by Remark 4.7 (i) the condition D = 0 yields e2 + 4bh = 0 and we obtain b = −e2/(4h). In this case for systems (4.31) we calculate: T =− 3e2(e2 + f2h)x2(ehx2 + 2fhxy − ey2)2/(16h), E1 =− e2f(e2 + f2h)/(8h), (4.36) and since T < 0, the condition E1 = 0 is equivalent to f = 0. By Remark 4.7 (ii) we deduce that in this case the singularities located outside the invariant line coalesced. So in the case E1 ̸= 0 the configuration (30) a, a, sn;S leads to the phase portrait given by Picture SIV .11 (see Figure 1). If E1 = 0 we have the topological configuration (34) a, a, cp;S which leads to the Picture SIV .12. Corresponding examples are the following: Picture SIV .11: [Ex: (0, 0, 0, 0, 1/2, 0), (−1, 2,−1,−1, 0, 1)]; Picture SIV .12: [Ex: (0, 0, 0, 0, 1/2, 0), (−1,−2, 0,−1, 0, 1)]. 1.3.1.1.2: B2 = 0. Then by Remark 4.7 we have b = 0 and in this case we obtain: T = −3e2f2x2y2(fhx− ey)2, E1 = −e2f3/2, (4.37) and evidently the condition T ̸= 0 implies E1 ̸= 0. So in this case we could only have the configuration (30) a, a, sn;S. Taking into account Remark 4.7 (ii) we arrive at Picture SIV .13: [Ex: (0, 0, 0, 0, 1/2, 0), (0,−1,−1,−1, 0, 1)]. 1.3.1.1.3:Ũ1 = 0. By Remark 4.7 we have b = f2/4 and in this case we obtain: T = −3f2(e2+f2h)y2(−fhx2+2exy+fy2)2/16, E1 = f3(e2+f2h)/16. (4.38) Clearly the condition T ̸= 0 implies E1 ̸= 0 and again we could have only the con- figuration (30) a, a, sn;S. In this case according to Remark 4.7 (ii) the singularities located on the invariant line coalesced. Therefore we arrive at Picture SIV .14: [Ex: (0, 0, 0, 0, 1/2, 0), (1/4, 1,−1,−1, 0, 1)]. 1.3.1.2: µ0 > 0. In this case by [4, Diagram 1, page 4] we could either have the topological configuration (37) s, a, sn;N if E1 ̸= 0, or (40) s, a, cp;N if E1 = 0 and ¬(C8), or (41) s, c, cp;N if E1 = 0 and (C8). However by Lemma 4.2 the conditions (C8) are incompatible with systems (4.31). So it remains to examine the phase portraits given by the topological configurations (37) and (40). EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 31 1.3.1.2.1: B2Ũ1 ̸= 0. Then by Remark 4.7 (i) we have e2 + 4bh = 0, i.e. b = −e2/(4h) and we calculate: T = −3e2(e2 + f2h)x2(ehx2 + 2fhxy − ey2)2/(16h), E1 = −e2f(e2 + f2h)/(8h), and since T < 0, the condition E1 = 0 is equivalent to f = 0. However for f = 0 we obtain T = −3e6x2(hx2− y2)2/(16h) and therefore the condition T < 0 implies h > 0, and this contradicts µ0 = −h > 0. Thus in the case B2Ũ1 ̸= 0 we could only have the configuration (37) s, a, sn;N and considering Remark 4.7 (ii) there are two singularities (saddle and node) on the invariant line x = 0 and a saddle-node outside. In the previous case with the configuration (8) s, s, a, a;N we have proved (see Lemma 4.4) that systems (4.31) could only have one phase portrait which is topo- logically equivalent to Picture SIV .4. We proved Lemma 4.4 by means of a deep study of slopes of the flow on several lines and isoclines and the slopes of some eigenvectors. Perhaps we could use the same technique here, but the existence of a saddle-node would complicate essentially such kind of proof because of the different properties of its eigenvectors. So we will give here another proof of the topological possibilities of the phase portraits and their perturbations. Lemma 4.8. Systems (4.31) with configuration (37) s, a, sn;N possess only one phase portrait which is topologically equivalent to Picture SIV .15. Proof. Even though systems (4.31) with configuration (8) s, s, a, a;N have a triple node at infinity, this node behaves topologically as a simple node, and therefore the configuration (8) is structurally stable for systems (4.31). We have proved in Lemma 4.4 that the only phase portrait is Picture SIV .4 which is the same as phase portrait S23,4 from [1]. So we have proved that the phase portrait could not coincide with any other four structurally stable phase portraits with configuration (8) given in [1] and it could neither have separatrix connections. In the case of configuration (37) s, a, sn;N for systems (4.31), again the triple infinite node behaves topologically as a simple node. Inside systems (4.31) the configuration (37) (corresponding to the condition D = 0) is part of the border of configuration (8) (corresponding to the condition D > 0). So since Picture SIV .4 is structurally stable, all the phase portraits with configuration (37) must be unstable of codimension one. According to [2] there are 9 codimension one structurally unstable phase portraits ranging from U1 A,2 to U1 A,10 with configuration (37). We do not plot them to save space. From these 9 phase portraits the only one which can be perturbed into phase portrait S23,4 inside the family (4.31) (having 4 finite singular points) is the phase portrait U1 A,7 from [2]. Since we have proved that only S23,4 is realizable with configuration (8), then the only codimension one realizable phase portrait for these systems is U1 A,7 which is topologically equivalent Picture SIV .15. For the same reason, as we proved that S23,4 was the only possibility for configura- tion (8) in QSAb, here we cannot have other phase portraits with separatrix connec- tions and hence U1 A,7 is also the only possibility for configuration (37). As an exam- ple of Picture SIV .15 we may take [Ex: (0, 0, 0, 0, 1/2, 0), (1/8, 1,−1, 2, 0, 1)]. □ 1.3.1.2.2: B2 = 0. According to (4.37) in this case the condition T ̸= 0 im- plies E1 ̸= 0, and we could only have the configuration (37) s, a, sn;N . Taking into 32 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 account Remark 4.7 (ii) we arrive at Picture (a) of Figure 4: [Ex: (0, 0, 0, 0, 1/2, 0), (0,−1,−1, 1, 0, 1)]. 1.3.1.2.3: Ũ1 = 0. Considering (4.38) we conclude that the condition T ̸= 0 implies E1 ̸= 0 and again systems (4.31) could only have the configuration (37) s, a, sn;N . In this case taking into account Remark 4.7 (ii) we obtain Picture (b) of Figure 4: [Ex: (0, 0, 0, 0, 1/2, 0), (1/4, 1,−1, 1/2, 0, 1)]. We remark that the phase portraits Picture (a) and Picture (b) from Figure 4 are topologically equivalent to Picture SIV .15. Figure 4. Some phase portraits of quadratic systems with η = M̃ = 0, C2 ̸= 0. 1.3.2: T > 0. According to [4, Diagram 1, page 4] systems (4.31) possess one real (double) and two complex singularities. Moreover in this case we could either have the topological configuration (44) sn;N if E1 ̸= 0, or (47) cp;N if E1 = 0. According to Remark 4.7 (iii) the condition B2 = 0 implies T < 0 and therefore we examine two cases: Ũ1 ̸= 0 and Ũ1 = 0. 1.3.2.1: Ũ1 ̸= 0. Then the condition D = 0 gives b = −e2/(4h) and we obtain the values of T and E1 given in (4.36). Clearly the condition T > 0 implies e2h(e2 + f2h) < 0, and then the condition E1 = 0 is equivalent to f = 0. So in the case E1 ̸= 0 we obtain Picture (c) of Figure 4: [Ex: (0, 0, 0, 0, 1/2, 0), (1,−2,−1, 1, 0, 1)], which is topologically equivalent to Picture SIV .16. If E1 = 0 we have a cusp and this leads to the phase portrait given in Picture SIV .17: [Ex: (0, 0, 0, 0, 1/2, 0), (1,−2, 0, 1, 0, 1)]. 1.3.2.2: Ũ1 = 0. Then b = f2/4 and in this case we obtain the values ofT and E1 given in (4.38). Evidently the condition T > 0 implies E1 ̸= 0 and we could have only the configuration (44) sn;N . Then we obtain a phase portrait topologically equivalent to Picture SIV .16: [Ex: (0, 0, 0, 0, 1/2, 0), (1/4, 0,−1, 1, 0, 1)]. 1.3.3:T = 0. Since D = 0, according to [4, Diagram 1, page 5] we consider two cases: P ̸= 0 and P = 0. 1.3.3.1: P ̸= 0. Then by [5, Table 6.2] systems (4.31) possess two double finite singularities, which are real if PR > 0 and complex if PR < 0. However we have the next lemma. Lemma 4.9. For a system (4.31) the conditions D = 0 = T and P ̸= 0 imply PR > 0 and T4B2 ̸= 0. EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 33 Proof. Suppose first that the condition B2 = 0. Then b = 0 and for systems (4.31) we have D = 0, T = −3e2f2x2y2(fhx− ey)2, P = e2f2x2y2, and clearly the condition P ̸= 0 implies T ̸= 0, i.e. we obtain a contradiction. So B2 ̸= 0 and then the condition D = −48b2(f2 − 4b)(e2 + 4bh) = 0 gives (f2 − 4b)(e2 + 4bh) = 0. We claim that the condition D = T = 0 ̸= P implies f2 − 4b = e2 + 4bh = 0. Indeed, assuming b = f2/4 we obtain: D = 0, T = −3(e2 + f2h)y2P, P = f2(fhx2 − 2exy − fy2)2/16, and therefore the conditions T = 0 and P ̸= 0 imply e2 + f2h = 0 and f ̸= 0. So we have h = −e2/f2 and this implies e2 + 4bh = 0. In a similar way could be shown that the condition e2 + 4bh = 0 leads to the condition f2 − 4b = 0 in the case D = T = 0 ̸= PB2 and we deduce that the claim is proved. On the other hand for b = f2/4 and h = −e2/f2 calculations yield D = T = 0, P = (ex+ fy)4/16, R = e2(ex+ fy)2/f2, T4 = e2f2/4. We observe that PR > 0 and T4 ̸= 0 and this completes the proof. □ Considering the conditions D = T = 0, PR > 0 and T4 ̸= 0, according to [4, Diagram 1, page 5] we arrive at the unique topological configuration (50) sn, sn;N . According to Remark 4.7 (ii) we have one saddle-node on the in- variant line and another outside. As a result we arrive at Picture SIV .18: [Ex: (0, 0, 0, 0, 1/2, 0), (1/4, 0,−1, 1, 0, 1)]. There are other topologically different phase portraits with two finite saddle-nodes and one infinite node as it is pointed out in [6] but in this case, the existence of the invariant straight line, or simply using continuity from the cases already studied s, a, sn;N and s, s, a, a;N , give that there is only one possible phase portrait in these conditions. 1.3.3.2: P = 0. We prove the following lemma. Lemma 4.10. Assume that for a system (4.31) the conditions µ0 ̸= 0 and D = T = P = 0 hold. Then the following conditions are equivalent: R ̸= 0, T4 ̸= 0 ⇔ b = e = 0, f ̸= 0; R ̸= 0, T4 = 0 ⇔ b = f = 0, e ̸= 0; R = 0 ⇔ b = e = f = 0. Moreover in the case R ̸= 0 this system possesses one simple and one triple real singularities, whereas for R = 0 it possess one singularity of multiplicity four. Proof. First we note that the number and multiplicities of the singular points of systems (4.31) follows directly from [5, Diagram 6.1]. Assume now that the condition D = 0 is fulfilled. Then we obtain b(f2−4b)(e2+ 4bh) = 0 and we consider all three cases given by this relation. (i) If b = 0 then for systems (4.31) we have T = −3e2f2x2y2(fhx− ey)2, P = e2f2x2y2, T4 = 2f4h. It is clear that the condition T = P = 0 implies ef = 0 and therefore R = 3f2h2x2 + 3e2y2. We observe that the condition f = 0 is equivalent to T4 = 0. Thus in the case R ̸= 0 we either have b = e = 0 and f ̸= 0 if T4 ̸= 0, or b = f = 0 and e ̸= 0 if T4 = 0. 34 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 (ii) Assuming b = f2/4 we obtain: T = −3(e2 + f2h)y2P, P = f2(fhx2 − 2exy − fy2)2/16, T4 = −f4h/4, and clearly the condition T = P = 0 yields f = 0 and we have T4 = 0 and R = 3e2y2. So if R ̸= 0 we obtain the conditions b = f = 0 and e ̸= 0. (iii) Suppose now that the condition b = −e2/(4h) holds. Then we calculate: T = −3h(e2 + f2h)x2P, P = e2(ehx2 + 2fhxy − ey2)2/(16h2), T4 = f2(9e2 + 8f2h)/4, and evidently the condition T = P = 0 gives e = 0 and in this case we obtain R = 3f2h2x2 and T4 = 2f4h. Therefore the condition R ̸= 0 implies T4 ̸= 0 and in this case we have b = e = 0 and f ̸= 0. It remains to observe that in all three cases (i), (ii) and (iii), the conditions T = P = R = 0 give b = e = f = 0 and this completes the proof. □ In what follows we consider each one of the subcases provided by Lemma 4.10. 1.3.3.2.1: R ̸= 0, T4 ̸= 0. By Lemma 4.10 we have b = e = 0 and considering [4, Diagram 1, page 6] we calculate: E3 = −f2h/4, T4 = 2f4h, µ0 = −h. If µ0 < 0 then h > 0 and this implies E3 < 0. Then by [4, Diagram 1, page 6] we arrive at the configuration (16) a, a;S. This leads to Picture SIV .6: [Ex: (0, 0, 0, 0, 1/2, 0), (0, 0, 1,−1, 0, 1)]. Assuming µ0 > 0 we obtain E3 > 0 and by the same Diagram 1 from [4] we obtain either the configuration (23) s, a;N if ¬(C10), or (24) s, c;N if (C10). However by Lemma 4.2 the conditions (C10) are not compatible with systems (4.31). Considering (4.33) we observe that if b = e = 0 both the triple and the simple singular points are located on the invariant line x = 0. So we obtain that the anti-saddle could not be a focus and hence the configuration (23) s, a;N leads to a phase portrait equivalent to Picture SIV .9: [Ex: (0, 0, 0, 0, 1/2, 0), (0, 0, 1, 1, 0, 1)]. 1.3.3.2.2: R ̸= 0, T4 = 0. By Lemma 4.10 we have b = f = 0 (this implies T3 = 0) and e ̸= 0. Therefore we obtain E3 = −e2/4 < 0 and following [4, Diagram 1, page 6] we need to distinguish two cases: µ0 < 0 and µ0 > 0. If µ0 < 0 then by [4, Diagram 1, page 5] we obtain either the configuration (59) a, es;S if ¬(C3), or (60) c, es;S if (C3). Considering the conditions (C3) from (3.1) in the case b = f = 0 we obtain: T4 = T3 = T2 = T1 = 0, σ = 3y ̸= 0,F = F1 = 0,H = −9h/2,B = −81e2/8 < 0. Since µ0 < 0 (i.e. h > 0) we have H < 0 and we deduce that the conditions (C3) are satisfied in the considered case. So we could have only the configuration (60) c, es;S which leads to Picture SIV .19: [Ex: (0, 0, 0, 0, 1/2, 0), (0, 1, 0,−1, 0, 1)]. Assume now µ0 > 0. Since E3 < 0 and T4 = T3 = 0 by [4, Diagram 1, page 5] we have the unique configuration (64) s, es;N which leads to Picture SIV .20: [Ex: (0, 0, 0, 0, 1/2, 0), (0, 1, 0, 1, 0, 1)]. 1.3.3.2.3: R = 0. By Lemma 4.10 we have b = e = f = 0 and this leads to the homogeneous quadratic systems ẋ = xy, ẏ = −hx2 + y2. EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 35 In this case by [4, Diagram 1, page 6] we could either have the topological config- uration (67) ee;S if µ0 < 0, or (47) hh;N if µ0 > 0. In the first case we arrive at Picture SIV .21: [Ex: (0, 0, 0, 0, 1/2, 0), (0, 0, 0, −1, 0, 1)]. The configuration (47) hh;N leads to a phase portrait topologically equivalent with Picture SIV .17: [Ex: (0, 0, 0, 0, 1/2, 0), (0, 0, 0, 1, 0, 1)]. 2: Case θ = Ñ = 0, B2 ̸= 0. Considering the systems (4.30) after renaming the parameters we shall examine the family: ẋ = y, ẏ = b+ ex+ fy + hx2. (4.39) For these systems calculations yield: C2 = −hx3, η = M̃ = 0, µ0 = µ1 = 0, µ2 = h2x2, U = h2(e2 − 4bh)x4y2, κ = K̃ = L̃ = 0, T4 = B1 = 0, σ = f. (4.40) Since µ0 = µ1 = 0 and the condition C2 ̸= 0 implies µ2 ̸= 0, according to [5, Table 6.2] the above systems possess finite singularities of total multiplicity two. More exactly we have the singularities M1,2(x1,2, 0), where x1,2 = ( − e± √ e2 − 4bh ) /(2h), and sign(e2 − 4bh) = sign(U). So we examine three subcases: U > 0, U < 0 and U = 0. 2.1: U > 0. Then considering (4.40) by [4, Diagram 3, page 9] we obtain either the configuration (23) s, a;N if ¬(C9), or (24) s, c;N if (C9). On the other hand comparing the conditions (C9) from (3.1) with (4.40) we deduce that all the conditions are satisfied except σ = 0, because for systems (4.39) we have σ = f . So we deduce that for this case the conditions (C9) are satisfied if and only if f = 0. Thus we obtain that the configuration (24) s, c;N leads to a phase portrait topologically equivalent with Picture SIV .10: [Ex: (0, 0, 1, 0, 0, 0), (0, 1, 0,−1, 0, 0)]. In the case of the configuration (23) s, a;N we arrive at the same possible phase portraits as in Figure 3, for which we confirm the existence of the Picture (a) topo- logically equivalent with Picture SIV .9: [Ex: (0, 0, 1, 0, 0, 0), (0, 1,−1,−1, 0, 0)]. Regarding the other two phase portraits given by Picture (b) and Picture (c) (see Figure 3), they cannot exist by a similar argument as we did in Lemma 4.6. Indeed, if f = 0 the systems are Hamiltonian, and so, they cannot have a focus. And if f ̸= 0, by Bendixson’s theorem (see [9, Theorem 7.10]) the divergence ∂P/∂x + ∂Q/∂y = f (for P and Q given in systems (4.39)) is strictly positive or negative in all the plane. Consequently these systems cannot have a periodic orbit. This proves that Picture (c) cannot be realizable when f ̸= 0. Moreover since the proof of Bendixson’s theorem also works for a loop formed by a singular point and a homoclinic orbit to it, it follows that in the case f ̸= 0 Picture (b) is also non realizable. 2.2: U < 0. By [4, Diagram 3, page 9] we obtain the topological configuration (12) N . This configuration leads to a phase portrait topologically equivalent with Picture SIV .5: [Ex: (0, 0, 1, 0, 0, 0), (−1, 0,−1,−1, 0, 0)]. 2.3: U = 0. In this case systems (4.39) possess a double singular point which could be a saddle-node or a cusp. But since for these systems the conditions κ = K̃ = L̃ = T4 = B1 = 0 hold, according to [4, Diagram 3, page 11] we arrive at 36 J. C. ARTÉS, J. LLIBRE, D. SCHLOMIUK N. VULPE EJDE-2024/50 the topological configuration (47) cp;N . This configuration leads to a phase portrait topologically equivalent with Picture SIV .17: [Ex: (0, 0, 1, 0, 0, 0), (−1, 2, 0,−1, 0, 0)]. Since all the cases are examined Theorem 4.1 is proved. □ 4.5.2. Systems (SV ): C2 = 0. These systems have the infinite line filled up with singularities and this family is considered in [18], where a total of 9 canonical forms of this family are presented: C2.1 – C2.9 (see Table 1, page 741). Directly from [18] and Theorem 1.2 we arrive at the next result. Theorem 4.11. Assume that for a quadratic system the condition C2 = 0 holds. Then this system belongs to the class QSAb if and only if the condition H2 10+H2 12 ̸= 0 is satisfied. Moreover its phase portrait is topologically equivalent to one of the pictures given in Figure 5 if and only the following corresponding conditions are verified: Picture C2.1 ⇔ H10 ̸= 0, H9 < 0; Picture C2.2(a) ⇔ H10 ̸= 0, H9 > 0, H7 ̸= 0; Picture C2.2(b) ⇔ H10 ̸= 0, H9 > 0, H7 = 0; Picture C2.3 ⇔ H10 ̸= 0, H9 = 0, H12 ̸= 0; Picture C2.4 ⇔ H10 ̸= 0, H9 = 0, H12 = 0; Picture C2.5(a) ⇔ H10 = 0, H12 ̸= 0, H11 > 0, µ2 < 0; Picture C2.5(b) ⇔ H10 = 0, H12 ̸= 0, H11 > 0, µ2 > 0; Picture C2.6 ⇔ H10 = 0, H12 ̸= 0, H11 < 0; Picture C2.7 ⇔ H10 = 0, H12 ̸= 0, H11 = 0. Figure 5. Global phase portraits of quadratic systems with C2 = 0. Acknowledgments. J. C. Artés and J. Llibre were partially supported by the Agencia Estatal de Investigación of Spain grants PID2022-136613NB-100 and the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2021SGR00113. J. Llibre is also partiaally supported by the Real Academia de Ciències i Arts de Barcelona. D. Schlomiuk was supported by NSERC Grant RN000355. N. Vulpe was supported by the Program SATGED 011303, Moldova State University and by the grant number 21.70105.31ŞD. EJDE-2024/50 DIFFERENTIAL SYSTEMS OF SECOND KIND 37 References [1] J. C. Artés, R. E. Kooij, J. Llibre; Structurally stable quadratic vector fields, Mem. Amer. Math. Soc., 134 (1998), 108 pp. [2] J. C. Artés, J. Llibre, A. C. Rezende; Structurally unstable quadratic vector fields of codi- mension one, Universitext, Birkhäuser, New York-Berlin, 2018, 283 pp. [3] J. C. Artés, J. Llibre, D. Schlomiuk, N. Vulpe; From topological to geometric equivalence in the classification of singularities at infinity for quadratic vector fields, Rocky Mountain J. of Math., 45 (2015), 29–113. [4] J. C. Artés, J. Llibre, D. Schlomiuk, N. I. Vulpe; Global topological configurations of singular- ities for the whole family of quadratic differential systems, Qualitative Theory of Dynamical Systems, 19(51) (2020) A [5] J. C. Artés, J. Llibre, D. 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Artés Departament de Matemátiques, Universitat Autónoma de Barcelona, 08193 Bellaterra, Barcelona, Spain Email address: joancarles.artes@uab.cat Jaume Llibre Departament de Matemátiques, Universitat Autónoma de Barcelona, 08193 Bellaterra, Barcelona, Spain Email address: jaume.llibre@uab.cat Dana Schlomiuk Département de Mathématiques et de Statistiques Université de Montréal, Montreal, Canada Email address: dana.schlomiuk@umontreal.ca Nicolae Vulpe Institute of Mathematics and Computer Science, Academy of Science of Moldova, 5 Academiei str, Chişinău, MD-2028, Moldova Email address: nvulpe@gmail.com 1. Introduction and statement of main results 2. Main invariant polynomials associated with the class QSAb 3. Preliminary results involving the use of polynomial invariants 4. Proof of Theorem 1.2 4.1. Subfamily defined by (A): >0 4.2. Subfamily defined by (B): <0 4.3. The subfamily defined by (C): =0, M"0365M=0 4.4. Subfamily defined by (D): = M"0365M=0 4.5. Phase portraits of the quadratic systems from the family (D) defined by Theorem 1.2 Acknowledgments References