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 equivalently $A^{(1)}_{N-1}$ invariant Painlev\'e equations. This construction identifies rational solutions with points on orbits of fundamental shift operators acting on first-order polynomial solutions derived for dressing chains of even periodicity. We also obtain conditions for the existence of special function solutions that occur for a special class of first-order polynomial solutions. For the special case of the N=4 dressing chain equations the method yields all the known rational solutions of Painlev\'e V equation. They are obtained through action of shift operators on the two independent first-order polynomial solutions. The formalism naturally extends to N=6 and beyond as shown in the paper.