  In this paper the two-dimensional Benney system describing long wave
propagation of a finite depth fluid motion and the multi-dimensional
Russo--Smereka kinetic equation describing a bubbly flow are considered. The
Hamiltonian approach established by J. Gibbons for one-dimensional Vlasov
kinetic equation is extended to a multi-dimensional case. A local Hamiltonian
structure associated with the hydrodynamic lattice of moments derived by D.J.
Benney is constructed. A relationship between this hydrodynamic lattice of
moments and the two-dimensional Vlasov kinetic equation is found. In the
two-dimensional case a Hamiltonian hydrodynamic lattice for the Russo--Smereka
kinetic model is constructed. Simple hydrodynamic reductions are presented.
