We extend Painlevé IV model by adding quadratic terms to its Hamiltonian obtaining two classes of models (coalescence and deformation) that interpolate between Painlevé IV and II equations for special limits of the underlying parameters. We derive the underlying Bäcklund transformations, symmetry structure and requirements to satisfy Painlevé property.

中文翻译：

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PainlevéIV和II方程的合并，变形和Bäcklund对称性

我们通过向其哈密顿量中添加二次项来扩展PainlevéIV模型，从而获得两类模型（聚结和变形），这些模型可以在PainlevéIV和II方程之间进行内插，以对基础参数进行特殊限制。我们推导了底层的Bäcklund变换，对称结构和满足Painlevé性质的要求。