Integrability of Liénard systems with a weak saddle

© 2016, Springer International Publishing. We characterize the local analytic integrability of weak saddles for complex Liénard systems, x˙ = y- F(x) , y˙ = ax, 0 ≠ a∈ 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 Liénard 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.