  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.
