Rational solutions of the Sasano systems of types $B_4^{(1)},$ $D_4^{(1)}$ and $D_5^{(2)}$

TL;DR

This paper classifies all rational solutions of certain Sasano systems related to affine Weyl groups, revealing their structure and symmetry properties through Bäcklund transformations.

Contribution

It provides a complete classification of rational solutions for specific Sasano systems associated with affine Weyl groups, expanding understanding of their solution space.

Findings

Rational solutions are classified into a single type by Bäcklund transformations.

The systems are expressed as coupled $P_{III}$ systems.

The classification reveals symmetry structures related to affine Weyl groups.

Abstract

We classify the rational solutions of the Sasano systems of types $B_4^{(1)},$ $D_4^{(1)}$ and $D_5^{(2)},$ which are all given by the coupled $P_{\rm III}$ systems and have the affine Weyl group symmetries of types $B_4^{(1)},$ $D_4^{(1)}$ and $D_5^{(2)}.$ The rational solutions are classified as one type by the B\"acklund transformation groups.