Abstract
Form-invariant Poisson brackets of hydrodynamic type with several spatial variables x1,…,xp are introduced for any commuting vector fields U1,…,Um on a manifold Mn. The incompressibility equation for certain parameters of the Poisson brackets is derived. Poisson brackets connected with an arbitrary dynamical system on Mn are constructed. A gauge transform for the Poisson brackets with one spatial variable x is defined and its action onto the tensor invariants is studied. A general identity between the coefficients bkij(u) and gij(u) is found.
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