  We consider the averaging of the weakly nonlocal Symplectic Structures
corresponding to local evolution PDE's in the Whitham method. The averaging
procedure gives the weakly nonlocal Symplectic Structure of Hydrodynamic Type
for the corresponding Whitham system. The procedure gives also the "action
variables" corresponding to the wave numbers of $m$-phase solutions of initial
system which give the additional conservation laws for the Whitham system.
