Coupled Painlev\'e systems with affine Weyl group symmetry of types $D_3^{(2)}$ and $D_5^{(2)}$

TL;DR

This paper introduces new coupled Painlevé systems with specific affine Weyl group symmetries, detailing their symmetries, holomorphy conditions, and invariant divisors, expanding the understanding of integrable systems.

Contribution

It presents novel coupled Painlevé systems with affine Weyl group symmetries of types D_3^{(2)} and D_5^{(2)}, including their symmetry and holomorphy conditions.

Findings

New two-parameter family of 4D coupled Painlevé systems with D_3^{(2)} symmetry

New four-parameter family of 8D coupled systems with D_5^{(2)} symmetry

Identification of symmetry, holomorphy conditions, and invariant divisors

Abstract

In this paper, we find a two-parameter family of coupled Painlev\'e systems in dimension four with affine Weyl group symmetry of type $D_3^{(2)}$. We also find a four-parameter family of 2-coupled $D_3^{(2)}$-systems in dimension eight with affine Weyl group symmetry of type $D_5^{(2)}$. We show that for each system, we give its symmetry and holomorphy conditions, respectively. These symmetries, holomorphy conditions and invariant divisors are new.