ar X iv : m at h - ph / 0 70 10 41 v 2 1 9 A pr 2 00 7 Coupled Painlevé VI system with E ( 1 ) 6 - symmetry

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. Introduction The Painlevé equations PJ (J = I, . . . ,VI) are ordinary differential equations of second order. It is known that these PJ admit the following affine Weyl group symmetries [O1]: PI PII PIII PIV PV PVI – A (1) 1 A (1) 1 ⊕A (1) 1 A (1) 2 A (1) 3 D (1) 4 Several extensions of the Painlevé equations have been studied from the viewpoint of affine Weyl group symmetry. The Noumi-Yamada system is a generalization of PII, PIV and PV for A (1) n -symmetry [NY1]. The coupled Painlevé VI system withD (1) 2n+2-symmetry is also studied [S]. In this paper, we present an new system of ordinary differential equations with E (1) 6 -symmetry. Our system can be expressed as a Hamiltonian system of sixth order with a coupled Painlevé VI Hamiltonian. In order to obtain this system, we consider a similarity reduction of a Drinfeld-Sokolov hierarchy of type E (1) 6 . The Drinfeld-Sokolov hierarchies