On the Lax pairs of the sixth Painleve' equation

TL;DR

This paper investigates the dependence of Lax pairs for the sixth Painleve' equation on its parameters, explaining why some Lax pairs exhibit meromorphic dependence and proposing ways to achieve holomorphic dependence.

Contribution

The paper analyzes the parameter dependence of known Lax pairs for PVI and suggests modifications to obtain holomorphic dependence on all parameters.

Findings

Fuchs Lax pair depends holomorphically on parameters

Jimbo-Miwa Lax pair shows meromorphic dependence on _

Proposed modifications can suppress meromorphic dependence

Abstract

The dependence of the sixth equation of Painleve' on its four parameters $(2 \alpha,-2 \beta,2 \gamma,1-2 \delta) =(\theta_{\infty}^2,\theta_{0}^2,\theta_{1}^2,\theta_{x}^2)$ is holomorphic, therefore one expects all its Lax pairs to display such a dependence. This is indeed the case of the second order scalar ``Lax'' pair of Fuchs, but the second order matrix Lax pair of Jimbo and Miwa presents a meromorphic dependence on $\theta_\infty$ (and a holomorphic dependence on the three other $\theta_j$). We analyze the reason for this feature and make suggestions to suppress it.