  We classify integrable Hamiltonian equations in 3D with the Hamiltonian
operator d/dx, where the Hamiltonian density h(u, w) is a function of two
variables: dependent variable u and the non-locality w such that w_x=u_y. Based
on the method of hydrodynamic reductions, the integrability conditions are
derived (in the form of an involutive PDE system for the Hamiltonian density
h). We show that the generic integrable density is expressed in terms of the
Weierstrass elliptic functions. Dispersionless Lax pairs, commuting flows and
dispersive deformations of the resulting equations are also discussed.
