Contact line bundles, foliations and integrability

We formulate the definition of the noncommutative integrability of contact systems on a contact manifold (M, H) using the Jacobi structure on the space of sections Γ(L) of a contact line bundle L. In the cooriented case, if the line bundle is trivial and H is the kernel of a globally defined contact form α, the Jacobi structure on the space of sections reduces to the standard Jacobi structure on (M,α). We therefore treat contact systems on cooriented and non-cooriented contact manifolds simultaneously. In particular, this allows us to work with dissipative Hamiltonian systems, where the Hamiltonian does not have to be preserved by the Reeb vector field. Bibliography: 32 titles.