On the analytic integrability of the Liénard analytic differential systems (Q896629)

The authors study the Liénard systems \[ \dot x=y, \quad \dot y=-g(x)-f(x)y \] with \(f\) and \(g\) being real analytic functions such that \(g(0)=0\), \(g(x)\not \equiv 0\), \(f(x)\not \equiv 0\). For such systems they prove a number of results on the existence of local analytic first integrals in a neighborhood of the origin and on the existence of global analytic first integrals.