  We formulate the constrained KP hierarchy (denoted by \cKP$_{K+1,M}$) as an
affine ${\widehat {sl}} (M+K+1)$ matrix integrable hierarchy generalizing the
Drinfeld-Sokolov hierarchy. Using an algebraic approach, including the graded
structure of the generalized Drinfeld-Sokolov hierarchy, we are able to find
several new universal results valid for the \cKP hierarchy. In particular, our
method yields a closed expression for the second bracket obtained through Dirac
reduction of any untwisted affine Kac-Moody current algebra. An explicit
example is given for the case ${\widehat {sl}} (M+K+1)$, for which a closed
expression for the general recursion operator is also obtained. We show how
isospectral flows are characterized and grouped according to the semisimple
{\em non-regular} element $E$ of $sl (M+K+1)$ and the content of the center of
the kernel of $E$.
