On the canonical forms of the multi-dimensional averaged Poisson brackets

We consider here special Poisson brackets given by the “averaging” of local multi-dimensional Poisson brackets in the Whitham method. For the brackets of this kind it is natural to ask about their canonical forms, which can be obtained after transformations preserving the “physical meaning” of the field variables. We show here that the averaged bracket can always be written in the canonical form after a transformation of “Hydrodynamic Type” in the case of absence of annihilators of initial bracket. However, in general case the situation is more complicated. As we show here, in more general case the averaged bracket can be transformed to a “pseudo-canonical” form under some special (“physical”) requirements on the initial bracket.