On the Lax pairs of the sixth Painleve' equation

Abstract

The dependence of the sixth equation of Painleve' on its four parameters $(2\hat{I}\pm ,-2{\hat{I}}^2,2{\hat{I}}^3,1-2\hat{I}´)=(\hat{I}{¸}_{\infty }^2,\hat{I}{¸}_0^2,\hat{I}{¸}_1^2,\hat{I}{¸}_x^2)$ 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 $\hat{I}{¸}_{\infty }$ (and a holomorphic dependence on the three other $\hat{I}{¸}_j$). We analyze the reason for this feature and make suggestions to suppress it.

6 pages, no figure, se'minaire franco-Japonais, Kyoto, 15-20 May 2006