Extended Symmetry of Higher Painlevé Equations of Even Periodicity and Their Rational Solutions

The structure of the extended affine Weyl symmetry group of higher Painlevé equations of N periodicity depends on whether N is even or odd. We find that for even N, the symmetry group (Formula presented.) contains the conventional Bäcklund transformations (Formula presented.), 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 the existence of degenerated solutions, and for (Formula presented.), we explicitly show how even reflection automorphisms cause degeneracy of a class of rational solutions obtained on the orbit of the translation operators of (Formula presented.). We obtain the closed expressions for the solutions and their degenerated counterparts in terms of the determinants of the Kummer polynomials.