  First, we give a brief review of the theory of the Lenard-Magri scheme for a
non-local bi-Poisson structure and of the theory of Dirac reduction. These
theories are used in the remainder of the paper to prove integrability of three
hierarchies of bi-Hamiltonian PDE's, obtained by Dirac reduction from some
generalized Drinfeld-Sokolov hierarchies.
