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 A ^ N − 1 ( 1 ) contains the conventional Bäcklund transformations s j , j = 1 , … , 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 the existence of degenerated solutions, and for N = 4 , we explicitly show how even reflection automorphisms cause degeneracy of a class of rational solutions obtained on the orbit of the translation operators of A ^ 3 ( 1 ) . We obtain the closed expressions for the solutions and their degenerated counterparts in terms of the determinants of the Kummer polynomials.