On the Lax pairs of symmetric Painleve equations

The symmetric forms of the Painlevé equations are a sequence of nonlinear dynamical systems in N+ 1 variables that admit the action of an extended affine Weyl group of type, as shown by Noumi and Yamada. They are equivalent to the periodic dressing chains studied by Veselov and Shabat, and by Adler. In this paper, a direct derivation of the symmetries of a corresponding sequence of (N+ 1) × (N+ 1) matrix linear systems (Lax pairs) is given. The action of the generators of the extended affine Weyl group of type on the associated Lax pairs is realized through a set of transformations of the eigenfunctions, and this extends to an action of the whole group.