Abstract. Generalized KdV hierarchies associated by Drinfeld-Sokolov reduction with grade 1 regular semisimple elements from non-equivalent Heisenberg subalgebras of a loop algebra G(X)C( lambda , lambda ^-1) are studied. The graded Heisenberg subalgebras containing such elements are labelled by the regular conjugacy classes in the Weyl group W(G) of the simple Lie algebra G. A representative w epsilon W(G) of a regular conjugacy class can be lifted to an inner automorphism of G given by w=exp (2i pi ad I₀/m), where I₀ is the defining vector of an sl₂ subalgebra of G. The grading is then defined by the operator d(m,I₀)=m lambda (d/d lambda )+ad I₀ and any grade 1 regular element Lambda from the Heisenberg subalgebra associated with (w) takes the form Lambda =(C₊+ lambda C_-), where (I₀, C_-)=-(m-1)C_- and C₊ is included in an sl₂ subalgebra containing I₀. The largest eigenvalue of adI₀ is (m-1) except for some cases in F₄, E_6,7,8. We explain how these Lie algebraic results follow from known results and apply them to construct integrable systems. If the largest ad I₀ eigenvalue is (m-1), then using any grade 1 regular element from the Heisenberg subalgebra associated with (w) we can construct a KdV system possessing the standard W-algebra defined by I₀ as its second Poisson bracket algebra. For G a classical Lie algebra, we derive pseudo-differential Lax operators for those non-principal KdV systems that can be obtained as discrete reductions of KdV systems related to gl_n. Non-Abelian Toda systems are also considered.

A corrigendum for this article has been published in 1996 J. Phys. A: Math. Gen. 29 1145

Print publication: Issue 20 (21 October 1995)