We provide a complete classification and an explicit representation of rational solutions to the fourth Painlevé equation ${\mathrm{P}}_{\mathrm{IV}}$ and its higher order generalizations known as the $A_{2n}$-Painlevé or Noumi-Yamada systems. The construction of solutions makes use of the theory of cyclic dressing chains of Schrödinger operators. Studying the local expansions of the solutions around their singularities we find that some coefficients in their Laurent expansion must vanish, which express precisely the conditions of trivial monodromy of the associated potentials. The characterization of trivial monodromy potentials with quadratic growth implies that all rational solutions can be expressed as Wronskian determinants of suitably chosen sequences of Hermite polynomials. The main classification result states that every rational solution to the $A_{2n}$-Painlevé system corresponds to a cycle of Maya diagrams, which can be indexed by an oddly coloured integer sequence. Finally, we establish the link with the standard approach to building rational solutions, based on applying Bäcklund transformations on seed solutions, by providing a representation for the symmetry group action on coloured sequences and Maya cycles.

我们为第四个Painlevé方程提供完整的分类和有理解的明确表示 ${\mathrm{P}}_{\mathrm{IV}}$ 及其较高阶的概括称为 ${\mathrm{一种}}_{\mathrm{2个}ñ}$-Painlevé或Noumi-Yamada系统。解决方案的构建利用了Schrödinger算子的循环修整链理论。通过研究解的奇异点周围的局部展开，我们发现它们的Laurent展开中的某些系数必须消失，这恰好表示了相关势的平凡单峰的条件。The characterization of trivial monodromy potentials with quadratic growth implies that all rational solutions can be expressed as Wronskian determinants of suitably chosen sequences of Hermite polynomials. 主要的分类结果表明，对于${\mathrm{一种}}_{\mathrm{2个}ñ}$-Painlevé系统对应于Maya图的循环，可以用奇数着色的整数序列进行索引。最后，通过提供对有色序列和Maya循环上对称组操作的表示形式，我们在将Bäcklund变换应用于种子解决方案的基础上，建立了与构建合理解决方案的标准方法的链接。