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We provide a complete classification and an explicit representation ofrational solutions to the fourth Painlev\'e equation PIV and its higher ordergeneralizations known as the $A_{2n}$-Painlev\'e or Noumi-Yamada systems. Theconstruction of solutions makes use of the theory of cyclic dressing chains ofSchr\"odinger operators. Studying the local expansions of the solutions aroundtheir singularities we find that some coefficients in their Laurent expansionmust vanish, which express precisely the conditions of trivial monodromy of theassociated potentials. The characterization of trivial monodromy potentialswith quadratic growth implies that all rational solutions can be expressed asWronskian determinants of suitably chosen sequences of Hermite polynomials. Themain classification result states that every rational solution to the$A_{2n}$-Painlev\'e system corresponds to a cycle of Maya diagrams, which canbe indexed by an oddly coloured integer sequence. Finally, we establish thelink with the standard approach to building rational solutions, based onapplying B\"acklund transformations on seed solutions, by providing arepresentation for the symmetry group action on coloured sequences and Mayacycles.

Complete classification of rational solutions of $A_{2n}$-Painlev\'{e} systems