Integrability Properties of Generalized Liénard Differential Equations

We classify generalized Liénard differential equations, sometimes called Liénard type equations, that have integrating factors of a special form. We present first integrals of the related equations. Our families of equations contain almost all known integrable generalized Liénard equations as partial cases. We find the necessary and sufficient conditions of the Liouvillian integrability for the polynomial generalized Liénard equations provided that they are nondegenerate near infinity. The degenerate near infinity polynomial generalized Liénard equations exist only under some restrictions on the degrees and leading coefficients of the polynomials parameterizing the equations. We provide a number of novel integrable polynomial and rational generalized Liénard equations. These equations arise in a variety of applications.