We consider a family of nonlinear oscillators with quadratic damping, that generalizes the Liénard equation. We show that certain nonlocal transformations preserve autonomous invariant curves of equations from this family. Thus, nonlocal transformations can be used for extending known classification of invariant curves to the whole equivalence class of the corresponding equation, which includes non-polynomial equations. Moreover, we demonstrate that an autonomous first integral for one of two non-locally related equations can be constructed in the parametric form from the general solution of the other equation. In order to illustrate our results, we construct two integrable subfamilies of the considered family of equations, that are non-locally equivalent to two equations from the Painlevé–Gambier classification. We also discuss several particular members of these subfamilies, including a traveling wave reduction of a nonlinear diffusion equation, and construct their invariant curves and first integrals.

我们考虑一系列具有二次阻尼的非线性振荡器，它推广了 Liénard 方程。我们表明，某些非局部变换保留了该族方程的自主不变曲线。因此，非局部变换可用于将已知的不变曲线分类扩展到相应方程的整个等价类，其中包括非多项式方程。此外，我们证明了两个非局部相关方程之一的自主第一积分可以从另一个方程的一般解以参数形式构造。为了说明我们的结果，我们构建了所考虑方程族的两个可积子族，它们非局部等效于来自 Painlevé-Gambier 分类的两个方程。