ar X iv : m at h - ph / 0 20 30 29 v 1 1 8 M ar 2 00 2 A new Lax pair for the sixth Painlevé equation associated with ŝo ( 8 )

In this article, we propose a new representation of Lax type for the sixth Painlevé equation. This representation, formulated in the framework of the loop algebra so(8)[z, z−1] of type D (1) 4 , provides a natural explanation of the affine Weyl group symmetry of PVI. After recalling a standard derivation of PVI, we describe in Section 2 fundamental Bäcklund transformations for PVI. In Section 3, we present our Lax pair for PVI associated with so(8)[z, z −1], and explain how the Bäcklund transformations arise from the linear problem. For the general background on Painlevé equations, we refer the reader to [2]. The authors would like to thank Professor Kanehisa Takasaki for valuable discussions in the early stage of this work.