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Integrability conditions for a class of cubic differential systems with a bundle of two invariant straight lines and one invariant cubic

Dimitru Cozma^a, Anatoli Dascalescu<sup>b

a</sup> Department of Mathematics, Tiraspol State University, 5 Gh. Iablocichin str., Chişinău, MD2069, Republic of Moldova
^b Institute of Mathematics and Computer Science, 5 Academiei str., Chişinău, MD2028, Republic of Moldova

Abstract: We determine conditions for the origin to be a center for a class of cubic differential systems having a bundle of two invariant straight lines and one invariant cubic. We prove that a fine focus $O(0,0)$ is a center if and only if the first three Lyapunov quantities vanish.

Keywords and phrases: cubic differential system, center-focus problem, invariant algebraic curve, integrability.

MSC: 34C05

Received: 26.03.2018

Language: English