Self-similar reductions of the Sawada-Kotera and Kupershmidt equations are studied. Results of Painlevé’s test for these equations are given. Lax pairs for solving the Cauchy problems to these nonlinear ordinary differential equations are found. Special solutions of the Sawada-Kotera and Kupershmidt equations expressed via the first Painlevé equation are presented. Exact solutions of the Sawada-Kotera and Kupershmidt equations by means of general solution for the first member of K₂ hierarchy are given. Special polynomials for expressions of rational solutions for the equations considered are introduced. The differential-difference equations for finding special polynomials corresponding to the Sawada-Kotera and Kupershmidt equations are found. Nonlinear differential equations of sixth order for special polynomials associated with the Sawada-Kotera and Kupershmidt equations are obtained. Lax pairs for nonlinear differential equations with special polynomials are presented. Rational solutions of the self-similar reductions for the Sawada-Kotera and Kupershmidt equations are given.

中文翻译：

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与Sawada-Kotera和Kupershmidt方程的自相似归约相关的Lax对和特殊多项式

研究了Sawada-Kotera和Kupershmidt方程的自相似约简。给出了这些方程式的Painlevé检验的结果。找到了求解这些非线性常微分方程的柯西问题的松散对。提出了通过第一个Painlevé方程表示的Sawada-Kotera和Kupershmidt方程的特殊解。借助K ₂的第一成员的一般解，Sawada-Kotera和Kupershmidt方程的精确解给出层次结构。介绍了用于表示所考虑方程式有理解的特殊多项式。找到了用于寻找与Sawada-Kotera和Kupershmidt方程相对应的特殊多项式的微分方程。获得了与Sawada-Kotera和Kupershmidt方程相关的特殊多项式的六阶非线性微分方程。给出了带有特殊多项式的非线性微分方程的松散对。给出了Sawada-Kotera和Kupershmidt方程自相似归约的有理解。