INTEGRABLE EVOLUTION-EQUATIONS ON ASSOCIATIVE ALGEBRAS

This paper surveys the classification of integrable evolution equation s whose field variables take values in an associative algebra, which i ncludes matrix, Clifford, and group algebra valued systems. A variety of new examples of integrable systems possessing higher order symmetri es are presented. Symmetry reductions lead to an associative algebra-v alued version of the Painleve transcendent equations. The basic theory of Hamiltonian structures for associative algebra-valued systems is d eveloped and the biHamiltonian structures for several examples are fou nd.