Lie Symmetry to second-order nonlinear differential equations and its first integrals

There are many well-known techniques for obtaining exact solutions of differential equations, but most of them are merely special cases of a few powerful symmetry methods. In this paper, we focus our attention on a second-order nonlinear ordinary differential equation of special forms with arbitrary parameters, which is a combination of Liénard-type equation and equation with quadratic friction. With the help of Lie Symmetry methods, we identify several integrable cases of this equation. And for each case, we use the Lie Symmetry method to derive the associated determining system, and apply it further to find infinitesimal generators under the given parametric conditions. After reducing them to canonical variables, we obtain a autonomous equation. Further, through the inverse transformations we identify the explicit first integrals form for each case