  We classify integrable third order equations in 2+1 dimensions which
generalize the examples of Kadomtsev-Petviashvili, Veselov-Novikov and Harry
Dym equations. Our approach is based on the observation that dispersionless
limits of integrable systems in 2+1 dimensions possess infinitely many
multi-phase solutions coming from the so-called hydrodynamic reductions.
%Conversely, the requirement of the existence of hydrodynamic reductions proves
to be an efficient classification criterion. In this paper we adopt a novel
perturbative approach to the classification problem. Based on the method of
hydrodynamic reductions, we first classify integrable quasilinear systems which
may (potentially) occur as dispersionless limits of soliton equations in 2+1
dimensions. To reconstruct dispersive deformations, we require that all
hydrodynamic reductions of the dispersionless limit are inherited by the
corresponding dispersive counterpart. This procedure leads to a complete list
of integrable third order equations, some of which are apparently new.
