On the quest for generalized Hamiltonian descriptions of 3D-flows generated by the curl of a vector potential

We examine Hamiltonian analysis of three-dimensional advection flow (x) over dot = v(x) of incompressible nature del. v= 0 assuming that the dynamics is generated by the curl of a vector potential v =del x A. More concretely, we elaborate Nambu-Hamiltonian and bi-Hamiltonian characters of such systems under the light of vanishing or non-vanishing of the quantity A.del x A. We present an example (satisfying A.del x A not equal 0) which can be written as in the form of Nambu-Hamiltonian and bi-Hamiltonian formulations. We present another example (satisfying A.del x A= 0) which we cannot able to write it in the form of a Nambu-Hamiltonian or bi-Hamiltonian system while it. can be manifested in terms of Hamiltonian one-form and yields generalized or vector Hamiltonian equations (x) over dot(i), = -epsilon(ijk )partial derivative eta(j)/partial derivative x(k).