Extended symmetry of higher Painlev\'e equations of even periodicity and their rational solutions

The structure of extended affine Weyl symmetry group of higher Painlev\'e equations of $N$ periodicity depends on whether $N$ is even or odd. We find that for even $N$, the symmetry group ${\widehat A}^{(1)}_{N-1}$ contains the conventional B\"acklund transformations $s_j, j=1,{\ldots},N$, the group of automorphisms consisting of cycling permutations but also reflections on a periodic circle of $N$ points, which is a novel feature uncovered in this paper. The presence of reflection automorphisms is connected to existence of degenerated solutions and for $N=4$ we explicitly show how the reflection automorphisms around even points cause degeneracy of a class of rational solutions obtained on the orbit of translation operators of ${\widehat A}^{(1)}_{3}$. We obtain the closed expressions for solutions and their degenerated counterparts in terms of determinants of Kummer polynomials.
Comment: 26 pages