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This paper provides a systematic description of the interplay between aspecific class of reductions denoted as \cKPrm ($r,m \geq 1$) of the primarycontinuum integrable system -- the Kadomtsev-Petviashvili ({\sf KP}) hierarchyand discrete multi-matrix models. The relevant integrable \cKPrm structure is ageneralization of the familiar $r$-reduction of the full {\sf KP} hierarchy tothe $SL(r)$ generalized KdV hierarchy ${\sf cKP}_{r,0}$. The important featureof \cKPrm hierarchies is the presence of a discrete symmetry structuregenerated by successive Darboux-B\"{a}cklund (DB) transformations. Thissymmetry allows for expressing the relevant tau-functions as Wronskians withina formalism which realizes the tau-functions as DB orbits of simple initialsolutions. In particular, it is shown that any DB orbit of a ${\sf cKP}_{r,1}$defines a generalized 2-dimensional Toda lattice structure. Furthermore, weconsider the class of truncated {\sf KP} hierarchies ({\sl i.e.}, those definedvia Wilson-Sato dressing operator with a finite truncated pseudo-differentialseries) and establish explicitly their close relationship with DB orbits of\cKPrm hierarchies. This construction is relevant for finding partitionfunctions of the discrete multi-matrix models.  The next important step involves the reformulation of the familiarnon-isospectral additional symmetries of the full {\sf KP} hierarchy so thattheir action on \cKPrm hierarchies becomes consistent with the constraints ofthe reduction. Moreover, we show that the correct modified additionalsymmetries are compatible with the discrete DB symmetry on the \cKPrm DBorbits.  The above technical arsenal is subsequently applied to obtain completeComment: LaTeX, 63 pg

Constrained KP Hierarchies: Additional Symmetries, Darboux–Bäcklund Solutions and Relations to Multi-Matrix Models