Holomorphic, rational Lax pairs of a $q$-discrete Painlevé VI equation

TL;DR

$q$-P$_{VI}$ is equipped with holomorphic, rational Lax pairs by replacing the diagonal, constant monodromy residue with a non-diagonal one, removing prior obstructions to holomorphy in the monodromy exponents. The authors derive fifteen rational candidate pairs for $(A_{1,11},A_{1,12})$ and present two explicit realizations, showing that the resulting ${ m q ext{-}P_{VI}}$ equations agree with the known JS1996 form up to a rescaling. This work extends holomorphic Lax representations from continuous ${ m P_{VI}}$ to the discrete setting and relates to moving-frame descriptions tied to Bonnet surfaces, potentially enabling discrete geometric interpretations. The results broaden the landscape of Lax representations for ${ m q ext{-}P_{VI}}$ and suggest avenues for connecting discrete Painlevé dynamics with discrete differential geometry.

Abstract

We build several matrix Lax pairs of ${\rm q-P_{\rm VI}}$ valid even when the two eigenvalues of the residue of the monodromy matrix at infinity are equal. Their elements are rational functions of the dependent variables.