  New approach in classification of integrable hydrodynamic chains is
established. This is the method of the Hamiltonian hydrodynamic reductions.
Simultaneously, this approach yields explicit Hamiltonian hydrodynamic
reductions of the Hamiltonian hydrodynamic chains. The concept of reducible
Poisson brackets is established. Also this approach is useful for
non-Hamiltonian hydrodynamic chains. The deformed Benney hydrodynamic chain is
considered.
