Why is my rational Painlevé V solution not unique?

Abstract

Under special conditions the Painlevé V equation has more than one rational solution solving it with the same parameters. In the setting of formalism that identifies points on orbits of the fundamental shift operators of $A_3^{(1)}$ affine Weyl group with rational solutions we derive conditions for such non-uniqueness to occur. We identify the seed solutions from which the non-unique solutions are generated and put forward a method to systematically obtain their closed expressions from the underlying seed solutions.

20 pages