The conservation of the Hamiltonian structures in the deformations of the Whitham systems

We consider the construction of the deformed Whitham system for the KdV equation in the one-phase case and investigate the conservation of the Hamiltonian properties in this situation. It is shown then that both the Gardner-Zakharov-Faddeev and the Magri brackets give the deformed Dubrovin-Novikov brackets (the brackets of Dubrovin-Zhang type) for the deformed Whitham system constructed by our procedure. The general approach used in the paper gives a scheme for the averaging of the Poisson structures in the general situation.