The averaging of multi-dimensional Poisson brackets for systems having   pseudo-phases
  We consider features of the Hamiltonian formulation of the Whitham method in the presence of pseudo-phases. As we show, an analog of the procedure of averaging of the Poisson bracket with the reduced number of the first integrals can be suggested in this case. The averaged bracket gives a Poisson structure for the corresponding Whitham system having the form similar to the structures arising in the presence of ordinary phases. 
A.Ya. Maltsev
nlin.SI