Generalized Drinfeld-Sokolov (DS) hierarchies are constructed through local reductions of Hamiltonian flows generated by monodromy invariants on the dual of a loop algebra. Following earlier work of De Groot et al, reductions based upon graded regular elements of arbitrary Heisenberg subalgebras are considered. We show that, in the case of the nontwisted loop algebra ℓ(gl n ), graded regular elements exist only in those Heisenberg subalgebras which correspond either to the partitions of n into the sum of equal numbers n = pr or to equal numbers plus one n = pr + 1. We prove that the reduction belonging to the grade 1 regular elements in the case n = pr yields the p × p matrix version of the Gelfand-Dickey r-KdV hierarchy, generalizing the scalar case p = 1 considered by DS. The methods of DS are utilized throughout the analysis, but formulating the reduction entirely within the Hamiltonian framework provided by the classical r-matrix approach leads to some simplifications even for p = 1.

Hamiltonian reduction is used to project a trivially integrable system on the Heisenberg double of SU (n, n), to obtain a system of Ruijsenaars type on a suitable quotient space. This system possesses BC n symmetry and is shown to be equivalent to the standard three-parameter BC n hyperbolic Sutherland model in the cotangent bundle limit.

Abstract. The p × p matrix version of the r-KdV hierarchy has been recently treated as the reduced system arising in a Drinfeld-Sokolov type Hamiltonian symmetry reduction applied to a Poisson submanifold in the dual of the Lie algebra gl pr ⊗ C I [λ, λ −1 ]. Here a series of extensions of this matrix Gelfand-Dickey system is derived by means of a generalized Drinfeld-Sokolov reduction defined for the Lie algebra gl pr+s ⊗ C I [λ, λ −1 ] using the natural embedding gl pr ⊂ gl pr+s for s any positive integer. The hierarchies obtained admit a description in terms of a p × p matrix pseudo-differential operator comprising an r-KdV type positive part and a non-trivial negative part. This system has been investigated previously in the p = 1 case as a constrained KP system.In this paper the previous results are considerably extended and a systematic study is presented on the basis of the Drinfeld-Sokolov approach that has the advantage that it leads to local Poisson brackets and makes clear the conformal (W-algebra) structures related to the KdV type hierarchies.Discrete reductions and modified versions of the extended r-KdV hierarchies are also discussed.

The Hamiltonian system given by H = ~p2 + V(q) with VEe 00 (Rn) is considered. A method for integrating such a system is that of separating the variables in the Hamilton-Jacobi equation. It is known that if such a separation is possible, then it can take place only when the equation is expressed in terms of generalized elliptic coordinates or in a degeneration of these. A criterion is proposed for deciding if separation is possible, and if it is, in which degeneration of elliptic coordinates it takes place.