Equivalence problem for compatible bi-Hamiltonian structures on three-dimensional orientable manifolds

We solve the equivalence problem for compatible bi-Hamiltonian structures on three-dimensional orientable manifolds via Cartan’s method of equivalence. The problem separates into two branches on total space, one of which ends up with the intransitive involutive structure equations. For the transitive case, we obtain an {e}-structure on both total and base spaces.