Algebraic invariant curves for the Liénard equation

Odani has shown that if deg g ≤ deg f then after deleting some trivial cases the polynomial system ẋ = y, ẏ = −f(x)y − g(x) does not have any algebraic invariant curve. Here we almost completely solve the problem of algebraic invariant curves and algebraic limit cycles of this system for all values of deg f and deg g. We give also a simple presentation of Yablonsky’s example of a quartic limit cycle in a quadratic system.