Coupled Painlev\'e systems in dimension four with affine Weyl group symmetry of types $A_4^{(2)}$ and $A_1^{(1)}$

TL;DR

This paper introduces new coupled Painlevé systems in four dimensions with specific affine Weyl group symmetries, detailing their symmetries, holomorphy conditions, and invariant divisors, and explores related partial differential systems.

Contribution

It presents novel coupled Painlevé systems with affine Weyl group symmetries of types A_4^{(2)} and A_1^{(1)}, including their symmetry and holomorphy conditions.

Findings

New coupled Painlevé systems with affine Weyl symmetries

Explicit symmetry and holomorphy conditions provided

Connection established between polynomial Hamiltonian system and autonomous systems

Abstract

We find a two-parameter family of coupled Painlev\'e systems in dimension four with affine Weyl group symmetry of type $A_4^{(2)}$. For a degenerate system of $A_4^{(2)}$ system, we also find a one-parameter family of coupled Painlev\'e systems in dimension four with affine Weyl group symmetry of type $A_1^{(1)}$. We show that for each system, we give its symmetry and holomorphy conditions. These symmetries, holomorphy conditions and invariant divisors are new. Moreover, we find a one-parameter family of partial differential systems in three variables with $W(A_1^{(1)})$-symmetry. We show the relation between its polynomial Hamiltonian system and an autonomous version of the system of type $A_1^{(1)}$.