  Toda lattice hierarchy and the associated matrix formulation of the
$2M$-boson KP hierarchies provide a framework for the Drinfeld-Sokolov
reduction scheme realized through Hamiltonian action within the second KP
Poisson bracket. By working with free currents, which abelianize the second KP
Hamiltonian structure, we are able to obtain an unified formalism for the
reduced $SL(M+1,M-k)$-KdV hierarchies interpolating between the ordinary KP and
KdV hierarchies. The corresponding Lax operators are given as superdeterminants
of graded $SL (M+1,M-k)$ matrices in the diagonal gauge and we describe their
bracket structure and field content. In particular, we provide explicit
free-field representations of the associated $W(M,M-k)$ Poisson bracket
algebras generalizing the familiar nonlinear $W_{M+1}$-algebra. Discrete
B\"{a}cklund transformations for $SL(M+1,M-k)$-KdV are generated naturally from
lattice translations in the underlying Toda-like hierarchy. As an application
we demonstrate the equivalence of the two-matrix string model to the $SL
(M+1,1)$-KdV hierarchy.
