Integrability of Liénard systems with a weak saddle

Abstract

We characterize the local analytic integrability of weak saddles for complex Lienard systems, \(\dot{x}=y-F(x),\)\(\dot{y}= ax\), \( 0\ne a\in \mathbb {C}\), with F analytic at 0 and \(F(0)=F'(0)=0.\) We prove that they are locally integrable at the origin if and only if F(x) is an even function. This result implies the well-known characterization of the centers for real Lienard systems. Our proof is based on finding the obstructions for the existence of a formal integral at the complex saddle, by computing the so-called resonant saddle quantities.