A TRANSVERSE STRUCTURE FOR THE LIE-POISSON BRACKET ON THE DUAL OF THE VIRASORO ALGEBRA

KdV equations can be described as Hamiltonian systems on the dual of the Virasoro algebra with the canonical Lie-Poisson (also called Berezin-Kirillov-Kostant) bracket. In this paper we give an explicit transverse structure for this Poisson manifold along a finite dimensional submanifold. The structure is linearizable and equivalent to the Lie-Poisson structure on s1(2, R)*. This problem is closely related to the classification of Hill's equations.