Classical Nambu brackets in higher dimensions

We consider n-linear Nambu brackets in dimension N higher than n. Starting from a Hamiltonian system with Poisson bracket and K Casimir invariants defined the phase space of = + 2M, where M is number effective degrees freedom, we investigate necessary sufficient condition for this to possess brackets. For case n 3, by looking possible solutions fundamental identity, found be 2, i.e., should have effectively one degree freedom. Locally, it shown that there only solution, up local change variables, solution canonical bracket, generated Levi–Cività tensors. These results generalize n(≥4)-linear