On the integrability conditions for a family of the Li\'enard-type equations

We study a family of Li\'enard--type equations. Such equations are used for the description of various processes in physics, mechanics and biology and also appear as traveling--wave reductions of some nonlinear partial differential equations. In this work we find new conditions for the integrability of this equations family. To this end we use an approach, which is bases on application of nonlocal transformations. By studying connections between this family of Li\'enard--type equations and type III Painlev\'e--Gambier equations, we obtain four new integrability criteria. We illustrate our results by providing examples of some integrable Li\'enard--type equations. We also discuss relationships between linearizability via nonlocal transformations of this family of Li\'enard--type equations and other integrability conditions for this family of equations.