Integrable systems of the intermediate long wave type in 2+1 dimensions

We classify 2 + 1 dimensional integrable systems with nonlocality of the intermediate long wave type. Links to the 2 + 1 dimensional waterbag system are established. Dimensional reductions of integrable systems constructed in this paper provide dispersive regularisations of hydrodynamic equations governing propagation of long nonlinear waves in a shear flow with piecewise linear velocity profile (for special values of vorticities).