Coupled Painlev\'e III systems with affine Weyl group symmetry of types $B_5^{(1)},D_5^{(1)}$ and $D_6^{(2)}$

TL;DR

This paper introduces four five-parameter families of six-dimensional coupled Painlevé III systems with specific affine Weyl group symmetries, providing explicit transformations and holomorphic characterizations.

Contribution

It presents new coupled Painlevé III systems with affine Weyl group symmetries of types $D_5^{(1)}$, $B_5^{(1)}$, and $D_6^{(2)}$, including explicit transformations and holomorphic characterizations.

Findings

Four families of coupled Painlevé III systems identified

Explicit birational and symplectic transformations established

Systems characterized by holomorphy conditions

Abstract

We find and study four kinds of five-parameter family of six-dimensional coupled Painlev\'e III systems with affine Weyl group symmetry of types $D_5^{(1)},B_5^{(1)}$ and $D_6^{(2)}$. We show that each system is equivalent by an explicit birational and symplectic transformation, respectively. We also show that we characterize each system from the viewpoint of holomorphy.