  We develop a theory of integrable dispersive deformations of 2+1 dimensional
Hamiltonian systems of hydrodynamic type following the scheme proposed by
Dubrovin and his collaborators in 1+1 dimensions. Our results show that the
multi-dimensional situation is far more rigid, and generic Hamiltonians are not
deformable. As an illustration we discuss a particular class of two-component
Hamiltonian systems, establishing the triviality of first order deformations
and classifying Hamiltonians possessing nontrivial deformations of the second
order.
