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

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.