Nonlinear oscillators described by polynomial Liénard differential equations arise in a variety of mathematical and physical applications. For a family of generalized Duffing–van der Pol oscillators we classify Darboux integrable cases and explicitly construct the corresponding generalized Darboux first integrals. We demonstrate that Darboux integrability is in strong correlation with the linearizability via the generalized Sundman transformations. We establish that the general solutions can be written in a parametric form. We prove that there are no limit cycles in integrable cases with autonomous Darboux first integrals.

中文翻译：

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二次-五次Duffing-van der Pol振荡器的Darboux第一积分和线性化

由多项式Liénard微分方程描述的非线性振荡器出现在各种数学和物理应用中。对于一类广义Duffing-van der Pol振荡器，我们对Darboux可积情况进行分类，并显式构造相应的广义Darboux第一积分。我们证明了Darboux可积性与通过广义Sundman变换的线性化高度相关。我们确定可以以参数形式编写一般解决方案。我们证明在具有自主Darboux第一积分的可积情况下没有极限环。