A new Lax pair for the sixth Painlev\'e equation associated with $\hat{\mathfrak{so}}(8)$

TL;DR

This paper constructs a novel Lax pair for the sixth Painlevé equation within the loop algebra e9b4f8(8), revealing the affine Weyl group symmetry as gauge transformations of the linear problem.

Contribution

It introduces a new Lax pair for P_{VI} based on e9b4f8(8) loop algebra, linking affine Weyl symmetry to gauge transformations.

Findings

New Lax pair for P_{VI} in e9b4f8(8) framework

Affine Weyl group symmetry as gauge transformations

Enhanced understanding of P_{VI} symmetries

Abstract

A new Lax pair for the sixth Painlev\'e equation $P_{VI}$ is constructed in the framework of the loop algebra $\mathfrak{so}(8)[z,z^{-1}]$. The whole affine Weyl group symmetry of $P_{VI}$ is interpreted as gauge transformations of the corresponding linear problem.