In this paper, the combination of Painlevé analysis and Lie group method is performed, the Painlevé properties (PPs) and Bäcklund transformations (BTs) of the nonlinear ${\varphi }^4$ types of equations are obtained under some conditions. Then, all of the point symmetries of the generalized ${\varphi }^4$ model are given by Lie group classification method. Furthermore, the exact solutions to the equations are investigated by the symmetry reductions and analytic method.

中文翻译：

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可积性质、李对称性分析和广义 φ4 模型的显式解

本文将 Painlevé 分析与李群方法相结合，对非线性的 Painlevé 性质（PPs）和 Bäcklund 变换（BTs）进行${\phi }^4$在某些条件下得到的方程类型。然后，广义的所有点对称${\phi }^4$模型由李群分类法给出。此外，通过对称简化和解析方法研究了方程的精确解。