On a conjecture on the integrability of Liénard systems

Abstract

We consider the Liénard differential systemsin C 2 where F (x) is an analytic function satisfying F (0) = 0 and F ′ (0) ̸ = 0. Then these systems have a strong saddle at the origin of coordinates. It has been conjecture that if such systems have an analytic first integral defined in a neighborhood of the origin, then the function F (x) is linear, i.e. F (x) = ax. Here we prove this conjecture, and show that when F (x) is linear and system (1) has an analytic first integral this is a polynomial.