Global Compatibility of Bi-Hamiltonian Structures on Three Dimensional Manifolds

It is shown in \cite{yazar6} that a 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, and the bi-Hamiltonian structure is globally compatible if and only if the Bott class of the complex codimension one foliation defined by the nonvanishing vector field vanishes. In this work, we constructed a dynamical system on $S^3$, which is globally bi-Hamiltonian, but the Hamiltonians are not globally compatible.