  In this paper we investigate multidimensional first-order quasi-linear
systems and find necessary conditions for them to admit Hamiltonian
formulation. The insufficiency of the conditions is related to the Poisson
cohomology of the admissible Hamiltonian operators. We present in detail the
example of two-dimensional, two-components systems of hydrodynamic type and of
a real reduction of the 3-waves system.
