Realizations of the Witt and Virasoro Algebras and Integrable Equations

In this paper we study realizations of infinite-dimensional Witt and Virasoro algebras. We obtain a complete description of realizations of the Witt algebra by Lie vector fields of first-order differential operators over the space ?(3). We prove that none of them admits non-trivial central extension, which means that there are no realizations of the Virasoro algebra in ?(3). We describe all inequivalent realizations of the direct sum of the Witt algebras by Lie vector fields over ?(3). This result enables complete description of all possible (1+1)- dimensional partial differential equations that admit infinite dimensional symmetry algebras isomorphic to the direct sum of Witt algebras. In this way we have constructed a number of new classes of nonlinear partial differential equations admitting infinite-dimensional Witt algebras. So new integrable models which admit infinite symmetry algebra are obtained.