Classical W-Algebras and Generalized Drinfeld-Sokolov Hierarchies for Minimal and Short Nilpotents

We derive explicit formulas for lambda-brackets of the affine classical -algebras attached to the minimal and short nilpotent elements of any simple Lie algebra . This is used to compute explicitly the first non-trivial PDE of the corresponding integrable generalized Drinfeld-Sokolov hierarchies. It turns out that a reduction of the equation corresponding to a short nilpotent is Svinolupov's equation attached to a simple Jordan algebra, while a reduction of the equation corresponding to a minimal nilpotent is an integrable Hamiltonian equation on 2h E double dagger-3 functions, where h E double dagger is the dual Coxeter number of . In the case when is both these equations coincide with the KdV equation. In the case when is not of type , we associate to the minimal nilpotent element of yet another generalized Drinfeld-Sokolov hierarchy.