Higher Gel'fand-Dikii structures

We apply the procedure of Magri and Weinstein to produce an infinity of compatible Poisson structures on a bihamiltonian manifold, to the case of the KdV phase space. The higher Gel'fand-Dikii structures thus obtained contain non local terms, which we express with the help of the r.h.s. of the KdV hierarchy. We also give a generating function for all these Poisson structues, in terms of the Baker-Akhiezer functions. Finally we describe the symplectic leaves of these Poisson structures.