Constrained KP Hierarchies: Darboux-B\"acklund Solutions and Additional Symmetries

We illustrate the basic notions of {\em additional non-isospectral symmetries} and their interplay with the discrete {\em \DB transformations} of integrable systems at the instance of {\em constrained Kadomtsev-Petviashvili} (\cKP) integrable hierarchies. As a main application we present the solution of discrete multi-matrix string models in terms of Wronskian $\t$-functions of graded $SL(m,1)$ \cKP hierarchies.