Nonstandard Drinfeld-Sokolov reduction

Subject to some conditions, the input data for the Drinfeld-Sokolov construction of KdV-type hierarchies is a quadruplet (A, Lambda, d(1), d(0)), where the d(i) are Z-gradations of a loop algebra A and Lambda is an element of A is a semisimple element of the nonzero d(1)-grade. A new sufficient condition on the quadruplet under which the construction works is proposed and examples are presented. The proposal relies on splitting the d(1)-grade zero part of A into a vector space direct sum of two subalgebras. This permits one to interpret certain Gelfand-Dickey-type systems associated with a nonstandard splitting of the algebra of pseudodifferential operators in the Drinfeld-Sokolov framework.