Bi-Hamiltonian structure in Frenet-Serret frame

We reduce the problem of constructing bi-Hamiltonian structure in three dimensions to the solutions of a Riccati equation in moving coordinates of Frenet-Serret frame. All explicitly constructed examples in the literature are exhausted by constant solutions. We conclude that in three dimensions vector fields which are not eigenvectors of the curl operator are locally bi-Hamiltonian.