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