  In this work, we show that an autonomous dynamical system defined by a
nonvanishing 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.

Symplectic Geometry