Abstract. Analogues of the KP and the Toda lattice hierarchy called dispersionless KP and Toda hierarchy are studied. Dressing operations in the dispersionless hierarchies are introduced as a canonical transformation, quantization of which is dressing operators of the ordinbry KP and Toda hierarchy. An alternative construction of general solutions of the ordinary KP and Toda hierarchy is given as twistor construction which is quatization of the similar construction of solutions of dispersionless hierarchies. These results as well as those obtained in previous papers are presented with proofs and necessary technical details.

An analogue of the KP hierarchy, the SDiff(2) KP hierarchy, related to the group of area-preserving diffeomorphisms on a cylinder is proposed. An improved Lax formalism of the KP hierarchy is shown to give a prototype of this new hierarchy. Two important potentials, S and τ , are introduced. The latter is a counterpart of the tau function of the ordinary KP hierarchy. A Riemann-Hilbert problem relative to the group of areadiffeomorphisms gives a twistor theoretical description (nonlinear graviton construction) of general solutions. A special family of solutions related to topological minimal models are identified in the framework of the Riemann-Hilbert problem. Further, infinitesimal symmetries of the hierarchy are constructed. At the level of the tau function, these symmetries obey anomalous commutation relations, hence leads to a central extension of the algebra of infinitesimal area-preserving diffeomorphisms (or of the associated Poisson algebra).

(2) group. Symmetries of the tau function turn out to have commutator anomalies, hencegive a representation of a central extension of the SDiff(2) algebra.

The eigenvectors of the Hamiltionians of the XYZ Gaudin model are constructed by means of the algebraic Bethe Ansatz. The construction is based on the quasi-classical limit of the corresponding results for the inhomogeneous higher spin eight vertex model. * EKS acknowledges the support of the Department of Mathematical Sciences, the University of Tokyo, where the most part of the work was done.