Using the only admissible rank-two realisations of the Lie algebra of the affine group in one dimension in terms of the Lie algebra of Lie symmetries of the Ermakov–Pinney (EP) equation, some classes of second order nonlinear ordinary differential equations solvable by reduction method are constructed. One class includes the standard EP equation as a special case. A new EP equation with a perturbed potential but admitting the same solution formula as EP itself arises. The solution of the dissipative EP equation is also discussed.

中文翻译：

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Ermakov-Pinney Lie代数的二维仿射子代数下的二阶常微分方程的不变性

根据Ermakov-Pinney（EP）方程的Lie对称性的Lie代数，使用一维仿射组的Lie代数的唯一允许的二阶实现，可以简化一些类的二阶非线性常微分方程方法被构造。一类包括标准EP方程作为特殊情况。出现了一个新的具有扰动电位但允许与EP本身相同的求解公式的EP方程。还讨论了耗散EP方程的解。