  Generalized Drinfeld-Sokolov (DS) hierarchies are constructed through local
reductions of Hamiltonian flows generated by monodromy invariants on the dual
of a loop algebra. Following earlier work of De Groot et al, reductions based
upon graded regular elements of arbitrary Heisenberg subalgebras are
considered. We show that, in the case of the nontwisted loop algebra
$\ell(gl_n)$, graded regular elements exist only in those Heisenberg
subalgebras which correspond either to the partitions of $n$ into the sum of
equal numbers $n=pr$ or to equal numbers plus one $n=pr+1$. We prove that the
reduction belonging to the grade $1$ regular elements in the case $n=pr$ yields
the $p\times p$ matrix version of the Gelfand-Dickey $r$-KdV hierarchy,
generalizing the scalar case $p=1$ considered by DS. The methods of DS are
utilized throughout the analysis, but formulating the reduction entirely within
the Hamiltonian framework provided by the classical r-matrix approach leads to
some simplifications even for $p=1$.
