Global Existence of Bi-Hamiltonian Structures on Orientable   Three-Dimensional Manifolds
  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. 
Melike I\c{s}im Efe, Ender Abado\u{g}lu
math.SG