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We develop a systematic approach to deriving rational solutions and obtainingclassification of their parameters for dressing chains of even N periodicity orequivalently $A^{(1)}_{N-1}$ invariant Painlev\'e equations. This constructionidentifies rational solutions with points on orbits of fundamental shiftoperators acting on first-order polynomial solutions derived for dressingchains of even periodicity. We also obtain conditions for the existence ofspecial function solutions that occur for a special class of first-orderpolynomial solutions. For the special case of the N=4 dressing chain equationsthe method yields all the known rational solutions of Painlev\'e V equation.They are obtained through action of shift operators on the two independentfirst-order polynomial solutions. The formalism naturally extends to N=6 andbeyond as shown in the paper.

On Rational Solutions of Dressing Chains of Even Periodicity