  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.

Exactly Solvable and Integrable Systems;Classical Analysis and ODEs