Darboux integrability of a cubic differential system with two parallel invariant straight lines

Abstract

In this paper we prove the Darboux integrability of a cubic differential system with a singular point of a center typer having at least two parallel invariant straight lines.

Real cubic differential systems with a linear center and multiple line at infinity

“…Consider the real cubic system of differential equations 3 ≡ 𝑝 (𝑥, 𝑦) , 𝑦 = −(𝑥 + 𝑔𝑥 2 + 𝑑𝑥𝑦 + 𝑏𝑦 2 + 𝑠𝑥 3 + 𝑞𝑥 2 𝑦 + 𝑛𝑥𝑦 2 + 𝑙 𝑦 3 ) ≡ 𝑞 (𝑥, 𝑦) , gcd( 𝑝, 𝑞) = 1, (𝑘, 𝑙, 𝑚, 𝑛, 𝑝, 𝑞, 𝑟, 𝑠) ≠ 0.…”

“…We suppose that at infinity system (1) has at most four distinct critical point, i.e. 𝑠𝑥 4 + (𝑘 + 𝑞)𝑥 3 𝑦 + (𝑚 + 𝑛)𝑥 2 𝑦 2 + (𝑙 + 𝑝)𝑥𝑦 3 + 𝑟 𝑦 4 0.…”

“…The homogeneous system associated to system (1) has the form 𝑥 = 𝑦𝑍 2 + (𝑎𝑥 2 + 𝑐𝑥𝑦 + 𝑓 𝑦 2 )𝑍 + 𝑘𝑥 3 + 𝑚𝑥 2 𝑦 + 𝑝𝑥𝑦 2 + 𝑟 𝑦 3 ≡ 𝑃 (𝑥, 𝑦, 𝑍) , 𝑦 = −(𝑥𝑍 2 + (𝑔𝑥 2 + 𝑑𝑥𝑦 + 𝑏𝑦 2 )𝑍 + 𝑠𝑥 3 + 𝑞𝑥 2 𝑦 + 𝑛𝑥𝑦 2 + 𝑙 𝑦 3 ) ≡ 𝑄 (𝑥, 𝑦, 𝑍) .…”

“…Some notions on multiplicity (algebraic, integrable, infinitesimal, geometric) of an invariant algebraic line and its equivalence for polynomial differential systems are given in [1]. Cubic differential systems with multiple invariant straight lines (including the line at infinity) were studied in [6], [13], [16], and the center problem for (1) with invariant straight lines was considered in [2], [3], [4], [5], [10], [12], [14], [15], [17].…”

“…4𝑘 5 𝑠 3 .If 𝜑 0 = 0 (respectively, {3𝑘3 − 𝑝𝑠 2 = 0, 𝜑 1 = 0}; {𝑘 3 − 2𝑘 𝑠 2 − 𝑝𝑠 2 = 0, 𝜑 1 = 0}), then Lemma 3.1, (73) (respectively, Lemma 3.1, (75); Lemma 3.1, (76)).System{𝜑 1 = 0, 𝜑 2 = 0} is incompatible because 𝑅𝑒𝑠𝑢𝑙𝑡𝑎𝑛𝑡 [𝜑 1 , 𝜑 2 , 𝑎] = 32𝑘 11 𝑠 8 ≠ 0.Under conditions (65) 𝐿 1 = 𝑔(2𝑎 − 𝑑). If 𝑔 = 0, then Lemma 3.1, (77), and if 𝑑 = 2𝑎, then Lemma 3.1, (78).Finally, when conditions (65) hold the first Lyapunov quantity is𝐿 1 = (𝑏𝑞 2 + 𝑑𝑞𝑠 − 2𝑏𝑠 2 )(𝑞 2 +𝑏 2 𝑠−𝑏𝑑𝑞).…”