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We derive explicit formulas for lambda-brackets of the affine classicalW-algebras attached to the minimal and short nilpotent elements of any simpleLie algebra g. This is used to compute explicitly the first non-trivial PDE ofthe corresponding intgerable generalized Drinfeld-Sokolov hierarchies. It turnsout that a reduction of the equation corresponding to a short nilpotent isSvinolupov's equation attached to a simple Jordan algebra, while a reduction ofthe equation corresponding to a minimal nilpotent is an integrable Hamiltonianequation on 2h-3 functions, where h is the dual Coxeter number of g. In thecase when g is sl_2 both these equations coincide with the KdV equation. In thecase when g is not of type C_n, we associate to the minimal nilpotent elementof g yet another generalized Drinfeld-Sokolov hierarchy.

Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents