Abstract. We study the Drinfeld-Sokolov hierarchies of type A ð1Þ n associated with the regular conjugacy classes of W ðA n Þ. A class of fourth order Painlevé systems is derived from them by similarity reductions.

A higher order Painlevé system of type D (1) 2n+2 was introduced by Y. Sasano. It is an extension of the sixth Painlevé equation (P VI ) for the affine Weyl group symmetry. It is also expressed as a Hamiltonian system of order 2n with a coupled Hamiltonian of P VI . In this paper, we discuss a derivation of this system from a Drinfeld-Sokolov hierarchy.

The sixth Painlevé equation arises from a Drinfeld-Sokolov hierarchy of type D 4 by similarity reduction.

We present an new system of ordinary differential equations with affine Weyl group symmetry of type E(1) 6 . This system is expressed as a Hamiltonian system of sixth order with a coupled Painlevé VI Hamiltonian.

The higher order Painlevé system of type D (1) 2n+2 was proposed by Y. Sasano as an extension of P VI for the affine Weyl group symmetry with the aid of algebraic geometry for Okamoto initial value space. In this article, we give it as the monodromy preserving deformation of a Fuchsian system.