Integrable lax hierarchies, their symmetry reductions and multi-matrix models

Some new developments in constrained Lax integrable systems and their applications to physics are reviewed. After summarizing the tau function construction of the K P hierarchy and the basic concepts of the symmetry of nonlinear equations, more recent ideas dealing with constrained K P models are described. A unifying approach to constrained K P hierarchy based on graded S L(r+n,n) algebra is presented and equivalence formulas are obtained for various pseudo-differential Lax operators appearing in this context. It is then shown how the Toda lattice structure emerges from constrained K P models via canonical Darboux-Baecklund transformations. These transformations enable to find simple Wronskian solutions for the underlying tau-functions. We also establish a relation between two-matrix models and constrained Toda lattice systems and derive from this relation expressions for the corresponding partition function. (author)