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

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