Foliations and Integrability
There is a close relationship, assuming everything is smooth, with vector fields: given a vector field on that is never zero, its integral curves will give a 1-dimensional foliation. (i.e. a codimension foliation).
This observation generalises to the Frobenius theorem, saying that the necessary and sufficient conditions for a distribution (i.e. an dimensional subbundle of the tangent bundle of a manifold) to be tangent to the leaves of a foliation, is that the set of vector fields tangent to the distribution are closed under Lie bracket. One can also phrase this differently, as a question of reduction of the structure group of the tangent bundle from to a reducible subgroup.
The conditions in the Frobenius theorem appear as integrability conditions; and the assertion is that if those are fulfilled the reduction can take place because local transition functions with the required block structure exist. For example, in the codimension 1 case, we can define the tangent bundle of the foliation as, for some (non-canonical) (i.e. a non-zero co-vector field). A given is integrable iff everywhere.
There is a global foliation theory, because topological constraints exist. For example in the surface case, an everywhere non-zero vector field can exist on an orientable compact surface only for the torus. This is a consequence of the Poincaré–Hopf index theorem, which shows the Euler characteristic will have to be 0. There are many deep connections with contact topology, which is the "opposite" concept.
Read more about this topic: Foliation