  Takhtajan has recently studied the consistency conditions for Nambu brackets,
and suggested that they have to be skew-symmetric, and satisfy Leibnitz rule
and the Fundamental Identity (FI, it is a generalization of the Jacobi
identity). If the n'th order Nambu brackets in dimension N is written in terms
of the Nambu tensor \eta, the FI implies two conditions on it, one algebraic
and one differential. The algebraic part of FI implies decomposability of \eta
and in this letter we show that the Nambu bracket can then be written in terms
of the usual totally antisymmetric n-dimensional tensor and n vector fields D.
Our main result is that the differential part of the FI is satisfied iff the
vector fields D commute. Examples are provided by integrable Hamiltonian
systems. It turns out that then the Nambu bracket itself guarantees that the
motions stays on the manifold defined by the constants of motion of the
integrable system, while the n-1 Nambu Hamiltonians determine the (possibly
non-integrable) motion on this manifold.
