Coupled Painlev\'e VI systems in dimension four with affine Weyl group symmetry of type $E_6^{(2)}$

TL;DR

This paper introduces a new four-parameter family of coupled Painlevé VI systems in four dimensions exhibiting affine Weyl group symmetry of type E6(2), expanding the understanding of higher order Painlevé systems.

Contribution

It presents the first example of higher order Painlevé systems with E6(2) symmetry, analyzing its symmetry and holomorphy conditions.

Findings

First example of E6(2)) higher order Painlevé system

Identifies symmetry and holomorphy conditions

Expands classification of Painlevé systems

Abstract

We find a four-parameter family of coupled Painlev\'e VI systems in dimension four with affine Weyl group symmetry of type $E_6^{(2)}$. This is the first example which gave higher order Painlev\'e type systems of type $E_{6}^{(2)}$. We study its symmetry and holomorphy conditions.