6 The sixth Painlevé equation arising from D ( 1 ) 4 hierarchy

The sixth Painlevé equation arises from a Drinfeld-Sokolov hierarchy of type D (1) 4 by similarity reduction. 2000 Mathematics Subject Classification: 34M55, 17B80, 37K10. Introduction The Drinfeld-Sokolov hierarchies are extensions of the KdV (or mKdV) hierarchy [DS]. It is known that their similarity reductions imply several Painlevé equations [AS, KK1, NY1]. For the sixth Painlevé equation (PVI), the relation with the A (1) 2 -type hierarchy is investigated [KK2]. On the other hand, PVI admits a group of symmetries which is isomorphic to the affine Weyl group of type D (1) 4 [O]. Also it is known that PVI is derived from the Lax pair associated with the algebra ŝo(8) [NY3]. However, the relation between D (1) 4 -type hierarchies and PVI has not been clarified. In this paper, we show that the sixth Painlevé equation is derived from a Drinfeld-Sokolov hierarchy of type D (1) 4 by similarity reduction. Consider a Fuchsian differential equation on P(C) dy dx2 + p1(x) dy dx + p2(x)y = 0, (0.1) with the Riemann scheme    x = t0 x = t1 x = t3 x = t4 x = λ x = ∞ 0 0 0 0 0 ρ θ0 θ1 θ3 θ4 2 ρ+ 1    ,