Integrability of C¹ invariant splittings

We derive some new conditions for integrability of dynamically defined C^1 invariant splittings in arbitrary dimension and co-dimension. In particular we prove that every 2-dimensional C^1 invariant decomposition on a 3-dimensional manifold satisfying a volume domination condition is uniquely integrable. In the special case of volume preserving diffeomorphisms we show that standard dynamical domination is already sufficient to guarantee unique integrability.