Loop algebra and visaoro symmetries of integrable hierarchies of KP type

We propose a systematic treatment of symmetries of KP integrable systems, including constrained (reduced) KP models cKP R, M (generalized AKNS hierarchies), and their multi component (matrix) generalizations. Any cKP R, M integrable hierarchy is shown to possess (Formula presented.) loop algebra (additional) symmetry. Also we provide a systematic construction of the full algebra of Virasoro additional symmetries in the case of constrained KP models which requires a non trivial modification of the known Orlov Schulman construction for the general unconstrained KP hierarchy. Multi component KP hierarchies are identified as ordinary (scalar) one component KP hierarchies supplemented with the Cartan subalgebra of the additional symmetry algebra, which provides the basis of a new method for construction of soliton like solutions. Davey Stewartson and N wave resonant systems arise as symmetry flows of ordinary CKPR, M hierarchies. © 2001, Taylor & Francis Group, LLC.