On the Lax pairs of the continuous and discrete sixth Painlevé equations

Abstract

Among the recently found discretizations of the sixth Painlevé equation P6, only the one of Jimbo and Sakai admits a discrete Lax pair, which does establish its integrability. However, a subtle restriction in this Lax pair prevents the possibility to generalize it in order to find the other missing Lax pairs. It happens that the same restriction already exists in the matrix Lax pair of Jimbo and Miwa for the continuous P6. In this preliminary article, we remove this last restriction and give a matrix Lax pair for P6 which is traceless, rational in the dependent and independent variables, holomorphic in the monodromy exponents, with four Fuchsian singularities in the complex plane of the spectral parameter. Its only minor drawback is the presence of the apparent singularity which always exists in the scalar Lax pair.