This is a preprint of: “Integrability of reversible and equivariant quadratic polynomial differential systems in the plane”, Jaume Llibre, Clàudia Valls, Rocky Mountain J. Math., vol. 49(2), 579–591, 2019. DOI: [10.1216/RMJ-2019-49-2-579] INTEGRABILITY OF REVERSIBLE AND EQUIVARIANT QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEMS IN THE PLANE JAUME LLIBRE AND CLAUDIA VALLS Abstract. We study the existence of first integrals for the class of reversible and equivariant quadratic polynomial differential systems in the plane. We put special emphasis in the study of the analytic first integrals. 1. Introduction For a planar differential system the existence of a first integral determines completely its phase portrait. However the explicit computation of first integrals is not an easy task. A first step is to compute those first integrals in different classes of functions such as the class of analytic functions. These last years quadratic vector fields have been investigated intensively as one of the easiest families of nonlinear differential systems, and more than one thousand papers have been published about these vector fields (see for instance [2, 8, 10, 9]), but the problem of classifying all the integrable quadratic vector fields remains open. For more information on the integrable differential vector fields in dimension 2, see for instance [3]. The reversible and the equivariant differential equations have symmetries and this is important because a symmetry of a differential equation is a transformation that sends solutions to solutions simplifying the study of these differential systems. The equations describing a physical or a bio- logical system often exhibit symmetries, therefore it is important to study such classes of systems. Reversible systems have an additional importance because reversibility has similar implications for the eigenvalues as in the Hamiltonian systems. Let φ : R2 → R2 be an involution, that is, φ ◦ φ = Id. We say that a polynomial vector field X is φ-reversible, if X satisfies Dφ(p)X(p) = −X ◦ φ(p), for all p ∈ R2 , and we say that X is φ-equivariant, if X satisfies Dφ(p)X(p) = X ◦ φ(p), for all p ∈ R2 . In this paper we characterize the existence of analytic first integrals for all φ-reversible and φ-equivariant quadratic polynomial vector fields in the plane. 2010 Mathematics Subject Classification. Primary: 37J35, 34Cxx. Key words and phrases. reversible, equivariant, quadratic planar polynomial equations, analytic first integral. 1 http://www.gsd.uab.cat 2 JAUME LLIBRE AND CLAUDIA VALLS It will be proved in section 2 that it is enough to consider the involution φ(x, y) = (x, −y). This result is not new (see for instance [6]) but since it is easy to prove we show it here for completeness. Since φ(x, y) = (x, −y) it is easy to obtain that the φ-reversible quadratic polynomial vector fields in R2 have the form x′ = y(a0 + a1 x), (1) y ′ = b0 + b1 x + b2 x2 + b3 y 2 , where a0 , a1 , b0 , b1 , b2 , b3 ∈ R, and that the φ-equivariant quadratic polyno- mial vector fields have the form x′ = a0 + a1 x + a2 x2 + a3 y 2 , (2) y ′ = (b0 + b1 x)y, where a0 , a1 , a2 , a3 , b0 , b1 ∈ R. Let U be an open subset of R2 such that R2 \U has zero Lebesgue measure. A non-constant function H : U → R is a first integral of the differential system (1) (respectively (2)) if it is constant on the solutions of system (1) (respectively (2)) contained in U . When a differential system has a first integral we say that it is integrable. We say that the first integral H is analytic when the function H is analytic in U and in this case the system is called analytically integrable. When U = R2 we say that H is a global analytic first integral. The main objective of this paper is to study the existence of analytic first integrals for systems (1) and (2). We shall see that system (1) is always analytically integrable, see Theorem 1. For system (2) we shall see that if a3 b1 (a1 b1 − 2a2 b0 ) = 0 then it is analytically integrable (see Theorems 2, 3 and 4). Finally in Theorem 5 we show that system (2) with a3 b1 (a1 b1 − 2a2 b0 ) ̸= 0 is not globally analytically integrable. Theorem 1. System (1) is analytically integrable. More precisely: (a) If a0 = a1 = 0, then a first integral is x. (b) If b0 = b1 = b2 = b3 = 0, then a first integral is y. In the rest of the theorem we assume that (a0 , a1 ) ̸= (0, 0) and that (b0 , b1 , b2 , b3 ) ̸= (0, 0, 0, 0). (c) If b3 = 0 and a1 = 0, then a first integral is x y2 − (6b0 + 3b1 x + 2b2 x2 ). (3) 3a0 (d) If b3 = 0 and a1 ̸= 0, then a first integral is x(2a1 b1 − 2a0 b2 + a1 b2 x) G0 y2 − − 3 log(a0 + a1 x), (4) a21 a1 where G0 = 2(a21 b0 − a0 a1 b1 + a20 b2 ). (e) If b3 ̸= 0 and a1 = 0, then a first integral is ( ) 2b3 ( 3 2 ) exp − x 2b3 y + a20 b2 + a0 b3 (b1 + 2b2 x) + 2b23 (b0 + b1 x + b2 x2 . (5) a0 INTEGRABILITY OF REVERSIBLE AND EQUIVARIANT SYSTEMS 3 (f) If b3 a1 ̸= 0 and (b3 − a1 )(2b3 − a1 ) ̸= 0, then a first integral is [ ] 2b − a3 2 G1 (a0 + a1 x) 1 y − , (6) b3 (b3 − a1 )(2b3 − a1 ) where G1 = −a21 b0 − a20 b2 − a0 b3 (b1 + 2b2 x) − 2b23 (b0 + b1 x + b2 x2 ) + a1 (a0 b1 + 3b0 b3 + 2b1 b3 x + b2 b3 x2 ). (g) If b3 a1 ̸= 0, b3 = a1 and b2 = 0, then a first integral is [ ] 1 2 a1 b0 + a0 b1 + 2a1 b1 x y + . (7) (a0 + a1 x)2 a21 (h) If b3 a1 ̸= 0, b3 = a1 and b2 ̸= 0, then a first integral is 1 [ 3 2 ] 2 a1 y − G2 − 2b2 (a0 + a1 x)2 log(a0 + a1 x) , (8) (a0 + a1 x) where G2 = 3a20 b2 − a21 (b0 + 2b1 x) − a0 a1 (b1 − 4b2 x). (i) If b3 a1 ̸= 0, b3 = a1 /2 and a1 b1 = 2a0 b2 , then a first integral is 1 [ 3 2 ] a1 y − 2(a20 b2 + a0 a1 b2 x + a21 (−b0 + b2 x2 )) . (9) a0 + a1 x (j) If b3 a1 ̸= 0, b3 = a1 /2 and a1 b1 ̸= 2a0 b2 , then a first integral is 1 [ 3 2 ] a1 y − G3 − 2(a1 b1 − 2a0 b2 )(a0 + a1 x) log(a0 + a1 x) , (10) a0 + a1 x where G3 = −2(a21 b0 − a0 a1 b1 + a20 b2 − a0 a1 b2 x − a21 b2 x2 ). Theorem 2. System (2) with condition a3 = 0 is analytically integrable. More precisely: (a) If a0 = a1 = a2 = 0, then a first integral is x. (b) If b0 = b1 = 0, then a first integral is y. In the rest of the theorem we assume that (a0 , a1 , a2 ) ̸= (0, 0, 0) and that (b0 , b1 ) ̸= (0, 0). (c) If a1 = a2 = 0, then a first integral is x log y − (2b0 + b1 x). (11) 2a0 (d) If a2 = 0 and a1 ̸= 0, then a first integral is a1 b0 − a0 b1 b1 log y − 2 log(a0 + a1 x) − x. (12) a1 a1 (e) If a2 ̸= 0 and a0 = a21 /(4a2 ), then a first integral is b1 2a2 b0 − a1 b1 log y − log(a1 + 2a2 x) + . (13) a2 a2 (a1 + 2a2 x) (f) If a2 ̸= 0 and a0 ̸= a21 /(4a2 ), then a first integral is ( ) b1 a1 b1 − 2a2 b0 a1 + 2a2 x log y − log(a0 + x(a1 + a2 x)) + √ arctan √ . 2a2 a2 4a0 a2 − a21 4a0 a2 − a21 (14) 4 JAUME LLIBRE AND CLAUDIA VALLS Theorem 3. System (2) with conditions a3 ̸= 0 and b1 = 0 is integrable with first integral y if b0 = 0, and √ √ (−a1 − 2a2 x)YS1 (S2 y) + a2 a3 y (YS1 +1 (S2 y) − YS1 −1 (S2 y)) √ √ , (15) (a1 + 2a2 x)JS1 (S2 y) + a2 a3 y (JS1 −1 (S2 y) − JS1 +1 (S2 y)) √ √ if b0 ̸= 0, where S1 = a21 − 4a0 a2 /(2b0 ), S2 = a2 a3 /b0 , Ya (z) is the Bessel function of the second kind, and Ja (z) is the Bessel function of the first kind. For more details on the Bessel functions see [1]. Theorem 4. System (2) with conditions a3 b1 ̸= 0, and a1 b1 = 2a2 b0 is analytically integrable. More precisely: (a) If a2 = 0, then a first integral is 2 (b0 + b1 x)2 − a3 b1 y 2 − (a0 b21 − a2 b20 ) log y. (16) b1 (b) If a2 = b1 , then a first integral is 1[ 2 2 2 2 2 ] a 0 b1 − a 2 b0 + b1 (b0 + b 1 x) − 2a 3 b1 y log y . (17) y2 (c) If a2 (a2 − b1 ) ̸= 0, then a first integral is − 2[ ] 2a y b1 a0 b1 (a2 − b1 ) + 2a2 b0 (a2 − b1 )x + a2 b1 (a2 − b1 )x2 + a2 a3 b1 y 2 . (18) Theorem 5. System (2) with a3 b1 (a1 b1 − 2a2 b0 ) ̸= 0 has no global analytic first integrals. Theorems 1, 2, 3, 4 and 5 are proved in sections 4, 5, 6, 7 and 8, respec- tively. 2. Reversible and equivariant quadratic polynomial vector fields We show in this section that in the definition of reversible and equivariant quadratic polynomial vector fields we can always consider that the involution φ is given by φ(x, y) = (x, −y). More precisely, we will prove the following result. Proposition 1. Let φ : R2 → R2 be a polynomial involution, and let X be a φ-reversible (respectively φ-equivariant) polynomial vector field of degree m > 1 in R2 . Then φ is conjugated to ψ given by ψ = diag (1, −1). Proof. We will prove the proposition only for the case in which X is a φ- reversible polynomial vector field, since the case in which X is φ-equivariant is completely analogous. We first show that under the assumptions, φ is linear. Let q be the degree of the polynomial involution φ : R2 → R2 . We will show that q = 1. Indeed since X is a φ-reversible vector field, we have that Dφ(p)X(p) = −X(φ(p)). This equation implies that if mq = q − 1 + m, or equivalently q(m − 1) = m − 1. Taking into account that m ̸= 1 we conclude that q = 1. Therefore, since φ is linear and it is an involution we get that det φ = ±1. By the Jordan’s normal form theorem, there is a linear change of variables h : R2 → R2 such that ψ = h−1 φh where ψ is formed by INTEGRABILITY OF REVERSIBLE AND EQUIVARIANT SYSTEMS 5 not zero eigenvalues λ1 and λ2 satisfying λ1 λ2 = ±1. Furthermore, using that ψ 2 = Id we conclude that λ1 = ±1 and λ2 = ±1. Hence we can always assume that λ1 = 1 and λ2 = −1. This concludes the proof.  3. Preliminary results The vector field X associated to the system x′ = P (x, y), y ′ = Q(x, y), (19) where P and Q are real polynomials in the variables x, y is defined by ∂ ∂ X = P (x, y) + Q(x, y) . ∂x ∂y We introduce the following auxiliary result due to Poincaré in [7], see also [4] for a direct proof. Through the paper Z+ will denote the set of non-negative integers. Theorem 6. Assume that the eigenvalues λ1 ̸= 0 and λ2 ̸= 0 at some singular point p of X do not satisfy any resonance condition of the form λ 1 k1 + λ 2 k2 = 0 for k1 , k2 ∈ Z+ with k1 + k2 > 0. Then system (19) has no analytic first integrals defined in a neighborhood of p. The following result is due to Li, Llibre and Zhang, see [5]. Theorem 7. Assume that the eigenvalues λ1 and λ2 at some singular point p of X satisfy that λ1 = 0 and λ2 ̸= 0. Then system (19) has no analytic first integrals in a neighborhood of p if p is isolated in the set of all singular points of X. 4. Proof of Theorem 1 It is clear that if a0 = a1 = 0 then x is a polynomial first integral, and if b0 = b1 = b2 = b3 = 0, then y is a polynomial first integral. So statements (a) and (b) are proved. From now on we assume that (a0 , a1 ) ̸= (0, 0) and (b0 , b1 , b2 , b3 ) ̸= (0, 0, 0, 0). We rewrite systems (1) into the form dy b0 + b1 x + b2 x2 b3 y = + . dx y(a0 + a1 x) a0 + a1 x Now taking the variable v = y 2 we get that the above system becomes the following linear differential system in the variable v, dv 2b3 2(b0 + b1 x + b2 x2 ) v′ = = v+ . (20) dx a0 + a1 x a0 + a1 x We consider eight cases which correspond with the eight statements (c)–(j) of the theorem. (c) b3 = 0 and a1 = 0. In this case the general solution of system (20) is x(6b0 + 3b1 x + 2b2 x2 ) v(x) = + H, 3a0 6 JAUME LLIBRE AND CLAUDIA VALLS where H is an integration constant. Hence system (1) under these assump- tions is integrable with the first integral (3). (d) b3 = 0 and a1 ̸= 0. In this case this case the general solution of system (20) is x(2a1 b1 − 2a0 b2 + a1 b2 x) G0 v(x) = + H + 3 log(a0 + a1 x), a21 a1 where H is an integration constant. Hence system (1) under these assump- tions is integrable with the first integral (4). (e) b3 ̸= 0 and a1 = 0. In this case the general solution of system (20) is ( ( 2b x ) ) 3 2b33 v(x) = −a20 b2 − a0 b3 (b1 + 2b2 x) − 2b23 b0 + x(b1 + b2 x) − b3 exp H , a0 where H is a first integral. Hence system (1) is integrable with the first integral (5). (f) b3 a1 ̸= 0 and (b3 − a1 )(2b3 − a1 ) ̸= 0. The general solution of system (20) is G1 v(x) = + (a0 + a1 x)2b3 /a1 H, b3 (b3 − a1 )(2b3 − a1 ) where H is a first integral. Hence system (1) has the first integral (6). (g) b3 a1 ̸= 0, b3 = a1 and b2 = 0. The general solution of system (20) is a1 b0 + a0 b1 + 2a1 b1 x v(x) = − + (a0 + a1 x)2 H. a21 Hence system (1) has the first integral (7). (h) b3 a1 ̸= 0, b3 = a1 and b2 ̸= 0. The general solution of system (20) is a31 v(x) = G2 + a31 (a0 + a1 x)2 H + 2b2 (a0 + a1 x)2 log(a0 + a1 x). Hence system (1) has the first integral (8). (i) b3 a1 ̸= 0, b3 = a1 /2 and a1 b1 = 2a0 b2 . In this case the general solution of system (2) is ( 2 ) a0 b2 + a0 a1 b2 x + a21 (−b0 + b2 x2 ) v(x) = 2 + (a0 + a1 x)H. a31 Hence system (1) has the first integral (9). (j) b3 a1 ̸= 0, b3 = a1 /2 and a1 b1 ̸= 2a0 b2 . In this case the general solution of system (20) is a31 v(x) = G3 + a31 (a0 + a1 x)H + 2(a1 b1 − 2a0 b2 )(a0 + a1 x) log(a0 + a1 x). Hence system (1) has the first integral (10). 5. Proof of Theorem 2 We consider system (2) with a3 = 0. If a0 = a1 = a2 = 0 then x is a polynomial first integral, and if b0 = b1 = 0 then y is a polynomial first integral. Therefore statements (a) and (b) hold. INTEGRABILITY OF REVERSIBLE AND EQUIVARIANT SYSTEMS 7 From now on we assume that (a0 , a1 , a2 ) ̸= (0, 0, 0) and that (b0 , b1 ) ̸= (0, 0). In this case we write system (2) in the form dx a0 + a1 x + a2 x2 = . (21) dy y(b0 + b1 x) It is clear that a first integral of system (21) is ∫ b0 + b1 x H = log y − dx. (22) a0 + a1 x + a2 x2 Now we consider four cases which correspond with the four statements (c)–(f) of the theorem. (c) a1 = a2 = 0. In this case, from system (22) we get that H given in (11) is a first integral. (d) a2 = 0 and a1 ̸= 0. Again from (22) we get that H given in (12) is a first integral. (e) a2 ̸= 0 and a0 = a21 /(4a2 ). From system (22) we obtain that H given in (13) is a first integral. (f) a2 ̸= 0 and a0 ̸= a21 /(4a2 ). In this case H given in (14) is a first integral. 6. Proof of Theorem 3 Here b1 = 0. When b0 = 0 it is clear that y is a polynomial first integral. Now we assume that b0 ̸= 0. In this case we write system (2) in the form dx a0 + a1 x + a2 x 2 + a3 y 2 x′ (y) = = . dy b0 y Solving it we get x(y) = N/D where √ N = − a2 a3 HyJS1 −1 (S2 y) − a1 HJS1 (S2 y) − a1 YS1 (S2 y) √ + a2 a3 y (HJS1 +1 (S2 y) − YS1 −1 (S2 y) + YS1 +1 (S2 y)) , D =2a2 (HJS1 (S2 y) + YS1 (S2 y)) , and where H is a first integral. Hence system (2) is integrable with the first integral (15). 7. Proof of Theorem 4 We introduce the change of variables (recall that a3 b1 ̸= 0) X − b0 X = b0 + b1 x, Y =y with inverse change x = , y = Y. b1 With this change of variables, system (2) becomes X ′ = A0 + A1 X + A2 X 2 + A3 Y 2 , (23) Y ′ = XY where a2 b20 2a2 b0 a2 A0 = a0 b1 − a1 b0 + , A1 = a1 − , A2 = , A3 = a3 b1 . b1 b1 b1 8 JAUME LLIBRE AND CLAUDIA VALLS Under the assumptions of the theorem, we can rewrite system (23) of the form dX A0 + A3 Y 2 A2 X = + , (24) dY XY Y where a0 b21 − a2 b20 a2 A0 = , A2 = , A3 = a3 b1 . b1 b1 We consider three cases which correspond with the three statements (a)–(c) of the theorem. (a) a2 = 0. In this case A2 = 0. Computing the general solution of system (24) and isolating the integration constant H we obtain H = X 2 − A3 Y 2 − 2A0 log Y, which in the original variables is the integral in (16). (b) a2 = b1 . In this case A2 = 1. Again computing the general solution of system (24) and isolating the integration constant H we obtain 1 [ ] H = 2 A0 + X 2 − 2A3 Y 2 log Y , Y which in the original variables is the integral in (17). (c) a2 (a2 − b1 ) ̸= 0. In this case we have A2 (A2 − 1) ̸= 0. The integration constant H of the general solution of system (24) is [ ] H = Y −2A2 − A0 + A0 A2 − A2 X 2 + A22 X 2 + A2 A3 Y 2 , which in the original variables is the integral in (18). 8. Proof of Theorem 5 We work with system (23). We consider four different cases. Case 1: A0 = 0. In this case the origin is a singular point. Its eigenvalues are 0 and A1 ̸= 0. Since this singular point is isolated (because A3 ̸= 0), Theorem 7 implies that system (23) has no analytic first integrals. Case 2: A0 ̸= 0 and A2 = 0. System (23) has the singular points ( √ ) ( √ ) ( ) A0 A0 A0 (X̄1 , Ȳ1 ) = 0, − , (X̄2 , Ȳ2 ) = 0, − − , (X̄3 , Ȳ3 ) = − , 0 . A3 A3 A1 The eigenvalues of (X̄3 , Ȳ3 ) are A1 and −A0 /A1 . Suppose that there exists k1 , k2 ∈ Z+ such that k1 λ1 + k2 λ2 = 0. Note that by Theorem 6 if such integers do not exist the theorem is proved. Then λ1 = −αλ2 with α a positive rational, and hence in particular λ1 λ2 = −αλ22 < 0. Since λ1 λ2 = −A0 we must have that A0 is positive. √ Moreover, the eigenvalues λ1 and λ2 of (X̄1 , Ȳ1 ) are (A1 ± A21 − 8A0 )/2. Again we must have that λ1 λ2 < 0, otherwise the theorem holds by Theorem 6. Note that λ1 λ2 = 2A0 > 0, and then we cannot have k1 λ1 + k2 λ2 = 0. The theorem is proved in this case. Case 3: A0 A2 ̸= 0 and A2 = A21 /(4A0 ). In this case (X̄, Ȳ ) = (−2A0 /A1 , 0) is a singular point. Its eigenvalues are 0 and −2A0 /A1 ̸= 0. Since this singular point is isolated (because A3 ̸= 0), Theorem 7 implies that system (23) has no analytic first integrals. INTEGRABILITY OF REVERSIBLE AND EQUIVARIANT SYSTEMS 9 Case 4: A0 A2 (A21 − 4A0 A2 ) ̸= 0. We assume that H = H(X, Y ) is a global analytic first integral in a neighborhood of Y = 0. We write it as ∑ H= Hk (X)Y k , (25) k≥0 where each Hk is an analytic function in the variable X. Without loss of generality we can assume that H has no constant term, i.e. H0 (0) = 0. We will show by induction that Hk = 0 for k ≥ 0. (26) Then clearly from (25) we will obtain that system (23) has no global analytic first integrals, and the proof of the theorem is done. Since H is a first integral of system (23) it must satisfy ∑ ∑ (A0 + A1 X + A2 X 2 + A5 Y 2 ) Hk′ (X)Y k + kXHk (X)Y k = 0. (27) k≥0 k≥0 Computing the terms of degree 0 in Y we get that H0′ (X) = 0, that is, H0 is a constant. Since H has no constant term we obtain that H0 = 0 which proves (26) for k = 0. ∑ for k = 0, . . . , ℓ − 1 and we will show it for Now assume that (26) is true k = ℓ. We have that H = Y ℓ k≥0 Hk+ℓ Y k . Computing the terms of degree ℓ in the variable Y in (27) we get that (A0 + A1 X + A2 X 2 )Hℓ′ (X) + ℓXHℓ (X) = 0. Solving this linear equation we get ( ( )) ℓ 2 − 2A2 A1 ℓ A1 + 2A2 X Hℓ (X) = K(A0 + A1 X + A2 X ) exp arctan , A2 à à √ where à = 4A0 A2 − A21 and K ∈ R. Since Hℓ must be a global analytic function in X, we get that K = 0 and thus Hℓ (X) = 0. This ends the induction process and the proof of the theorem is completed. Acknowledgments The first author has been supported by the grants MICIIN/FEDER MTM 2009-03437, AGAUR 2009SGR 410 and ICREA Academia. The second author has been partially supported by FCT through CAMGSD. References [1] M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with for- mulas, graphs, and mathematical tables, National Bureau of Standards Applied Math- ematics Series, 55, Washington, D.C., 1964. [2] L. Cairó and J. Llibre, Phase portraits of planar semi-homogeneous systems I, Nonlinear Anal. 29 (1997), 783–811. [3] J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, On the integrability of two-dimensional flows, J. 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Yanqian, Qualitative Theory of Polynomial Differential Systems, Shanghai Sci- entific & Technical Publishers, Shanghai, 1995 (in Chinese). Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, 1049-001 Lisboa, Portugal E-mail address: cvalls@math.ist.utl.pt