Abstract. We derive explicit formulas for λ-brackets of the affine classical Walgebras 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 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 2hˇ− 3 functions, where hˇis 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 Cn, we associate to the minimal nilpotent element of g yet another generalized Drinfeld-Sokolov hierarchy.