On a conjecture on the integrability of Liénard systems (Q2174827)

Local analytical integrability of the Liénard system \[ \dot x=y+F(x), \qquad \dot y= x, \qquad (1) \] where \(F(x)\) is an analytic function satisfying \(F(0) = 0\) and \(F'(0) \ne 0\), is investigated. First, the authors prove that if system (1) has a local analytic first integral defined in a neighborhood of the origin, then \(a = F'(0) = \pm (k_1 - k_2)/\sqrt{k_1 k_2},\) where \(a \ne 0\) and \(k_1\) and \(k_2\) are coprime positive integers. Then, it is proved that if system (1), with \(a\) defined above, has an analytic first integral in a neighborhood of the origin, then \(F(x) = ax\) and the system has a polynomial first integral of a certain form.