Are there any field of mathematics, except dynamical systems, where one needs to integrate continuous sub-bundles of the tangent space?

More specifically given a smooth manifold of $M$ and a continuous sub-bundle $E$ of $TM$, by integrating I mean finding a foliation whose leaves have tangent spaces that coincide with $E$ (this question also locally happens to be equivalent to solving a 1st order system of linear homogeneous PDE). In dynamical systems such sub-bundles arise as sub-spaces that are left invariant by differential of some diffeomorphism and sometimes being able to integrate them helps for certain classifications or statistical properties.

I am wondering if continuous sub-bundles OR continuous linear homogeneous systems of PDEs appear else where where it is important to know whether if you can integrate them or not?

Thanks