  In three dimensions, the construction of bi-Hamiltonian structure can be
reduced to the solutions of a Riccati equation with the arclength coordinate of
a Frenet-Serret frame being the independent variable. Explicit integration of
conserved quantities are connected with the coefficients of Riccati equation
which are elements of the third cohomology class. All explicitly constructed
examples of bi-Hamiltonian systems are exhausted when this class along with the
first one vanishes. The latter condition provides integrating factor for
explicit integration of Hamiltonian functions. For the Darboux-Halphen system,
the Godbillon-Vey invariant is shown to arise as obstruction to integrability
of integrating factor.
