On the integrability of Liénard systems with a strong saddle

We study the local analytic integrability for real Liénard systems, ẋ=y−F(x), ẏ=x, with F(0)=0 but F′(0)≠0, which implies that it has a strong saddle at the origin. First we prove that this problem is equivalent to study the local analytic integrability of the [p:−q] resonant saddles. This result implies that the local analytic integrability of a strong saddle is a hard problem and only partial results can be obtained. Nevertheless this equivalence gives a new method to compute the so-called resonant saddle quantities transforming the [p:−q] resonant saddle into a strong saddle.