On Rational Solutions of Dressing Chains of Even Periodicity

We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalent Painlevé equations invariant under AN−1(1) symmetry. This formalism identifies rational solutions (as well as special function solutions) with points on orbits of fundamental shift operators of AN−1(1) affine Weyl groups acting on seed configurations defined as first-order polynomial solutions of the underlying dressing chains. This approach clarifies the structure of rational solutions and establishes an explicit and systematic method towards their construction. For the special case of the N=4 dressing chain equations, the method yields all the known rational (and special function) solutions of the Painlevé V equation. The formalism naturally extends to N=6 and beyond as shown in the paper