Global existence of Bi-Hamiltonian structures on orientable three-dimensional manifolds

In this work, we show that an autonomous dynamical system defined by a non-vanishing vector field on an orientable three-dimensional manifold is globally bi-Hamiltonian if and only if the first Chern class of the normal bundle of the given vector field vanishes. Furthermore, the bi-Hamiltonian structure is globally compatible if and only if the Bott class of the complex codimension one foliation defined by the given vector field vanishes.