Self-similar reductions for equations of the Kupershmidt and Sawada – Kotera hierarchies are considered. Algorithms for constructing a Lax pair for equations of these hierarchies are presented. Lax pairs for ordinary differential equations of the fifth, seventh and eleventh orders corresponding to the Kupershmidt and the Sawada – Kotera hierarchies are given. The Lax pairs allow us to solve these equations by means of the inverse monodromy transform method. The application of the Painlevé test to the seventh order of the similarity reduction for the Kupershmidt hierarchy is demonstrated. It is shown that special solutions of the similarity reductions for the Kupershnmidt and Sawada – Kotera hierarchies are determined via the transcendents of the \(K_{1}\) and \(K_{2}\) hierarchies. Rational solutions of the similarity reductions of the modified Kupershmidt and Sawada – Kotera hierarchies are given. Special polynomials associated with the self-similar reductions of the Kupershmidt and Sawada – Kotera hierarchies are presented. Rational solutions of some hierarchies are calculated by means of the Miura transformations and taking into account special polynomials.

考虑了 Kupershmidt 和 Sawada-Kotera 层次结构方程的自相似约简。提出了为这些层次结构的方程构建 Lax 对的算法。给出了对应于 Kupershmidt 和 Sawada-Kotera 层次的五阶、七阶和十一阶常微分方程的松对。Lax 对允许我们通过逆单向变换方法来求解这些方程。演示了 Painlevé 测试在 Kupershmidt 层次结构相似度减少的七阶中的应用。结果表明，Kupershnmidt 和 Sawada-Kotera 层次的相似度约简的特殊解是通过\(K_{1}\)和\(K_{2}\)的超越确定的层次结构。给出了修正的 Kupershmidt 和 Sawada-Kotera 层次相似度约简的合理解。介绍了与 Kupershmidt 和 Sawada – Kotera 层次结构的自相似约简相关的特殊多项式。一些层次的有理解是通过 Miura 变换并考虑到特殊多项式来计算的。