On connections of the Liénard equation with some equations of Painlevé-Gambier type

The Liénard equation is used in various applications. Therefore, constructing general analytical solutions of this equation is an important problem. Here we study connections between the Liénard equation and some equations from the Painlevé-Gambier classification. We show that with the help of such connections one can construct general analytical solutions of the Liénard equation's subfamilies. In particular, we find three new integrable families of the Liénard equation. We also propose and discuss an approach for finding one-parameter families of closed-form analytical solutions of the Liénard equation.