  The $p\times p$ matrix version of the $r$-KdV hierarchy has been recently
treated as the reduced system arising in a Drinfeld-Sokolov type Hamiltonian
symmetry reduction applied to a Poisson submanifold in the dual of the Lie
algebra $\widehat{gl}_{pr}\otimes {\Complex}[\lambda, \lambda^{-1}]$. Here a
series of extensions of this matrix Gelfand-Dickey system is derived by means
of a generalized Drinfeld-Sokolov reduction defined for the Lie algebra
$\widehat{gl}_{pr+s}\otimes {\Complex}[\lambda,\lambda^{-1}]$ using the natural
embedding $gl_{pr}\subset gl_{pr+s}$ for $s$ any positive integer. The
hierarchies obtained admit a description in terms of a $p\times p$ matrix
pseudo-differential operator comprising an $r$-KdV type positive part and a
non-trivial negative part. This system has been investigated previously in the
$p=1$ case as a constrained KP system. In this paper the previous results are
considerably extended and a systematic study is presented on the basis of the
Drinfeld-Sokolov approach that has the advantage that it leads to local Poisson
brackets and makes clear the conformal ($\cal W$-algebra) structures related to
the KdV type hierarchies. Discrete reductions and modified versions of the
extended $r$-KdV hierarchies are also discussed.
