KdV equation on Riemann surfaces

We define a generalization of the KdV equation to Riemann surfaces, together with the corresponding hierarchy of equations and infinite set of charges in involution. We show that the second hamiltonian structure gives rise to a realization of the Krichever-Novikov algebra. Relying only on covariance, we define more general hamiltonian structures that give rise to generalizations to higher genus of some spin algebras. (orig.)