  This paper is devoted to the classification of integrable Davey-Stewartson
type equations. A list of potentially deformable dispersionless systems is
obtained through the requirement that such systems must be generated by a
polynomial dispersionless Lax pair. A perturbative approach based on the method
of hydrodynamic reductions is employed to recover the integrable systems along
with their Lax pairs. Some of the found systems seem to be new.
