Center conditions for a cubic system with two invariant straight lines and one invariant cubic

Abstract

In this paper we study the problem of the center for cubic system having three algebraic solutions l1 = 0, l2 = 0, Φ = 0 in generic position and prove the following theorem: Theorem 1. Let the cubic system have two invariant straight lines l1 = 0, l2 = 0 and one irreducible invariant cubic Φ = 0. Then a fine focus O(0, 0) is a center if and only if the first three Lyapunov quantities vanish.