Co-adjoint orbits of the generalised S1(2), S1(3) KdV hierarchies

Abstract In this paper we develop two co-adjoint orbit constructions for the phase spaces of the generalised Sl(2) and Sl(3) KdV hierarchies. This involves the construction of two group actions in terms of Yang-Baxter operators, and a hamiltonian reduction of the co-adjoint orbits. The Poisson brackets are reproduced by the Kirillov construction. From this construction we obtain a “natural” gauge fixing proceedure for the generalised hierarchies.