  We derive explicit formulas for lambda-brackets of the affine classical
W-algebras attached to the minimal and short nilpotent elements of any simple
Lie algebra g. This is used to compute explicitly the first non-trivial PDE of
the corresponding intgerable 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-3 functions, where h is the dual Coxeter number of g. In the
case when g is sl_2 both these equations coincide with the KdV equation. In the
case when g is not of type C_n, we associate to the minimal nilpotent element
of g yet another generalized Drinfeld-Sokolov hierarchy.
