Classical Affine {mathcal{W}}-Algebras for {mathfrak{gl}_N} and Associated Integrable Hamiltonian Hierarchies

We apply the new method for constructing integrable Hamiltonian hierarchies of Lax type equations developed in our previous paper to show that all ({mathcal{W}})-algebras ({mathcal{W}(mathfrak{gl}N, f)}) carry such a hierarchy. As an application, we show that all vector constrained KP hierarchies and their matrix generalizations are obtained from these hierarchies by Dirac reduction, which provides the former with a bi-Poisson structure.