On the infinite-dimensional symmetry group of the Davey–Stewartson equations

The Lie algebra of the group of point transformations, leaving the Davey–Stewartson equations (DSE’s) invariant, is obtained. The general element of this algebra depends on four arbitrary functions of time. The algebra is shown to have a loop structure, a property shared by the symmetry algebras of all known (2+1)-dimensional integrable nonlinear equations. Subalgebras of the symmetry algebra are classified and used to reduce the DSE’s to various equations involving only two independent variables.