Vessiot distribution

[Tehseen 2014]
For finite $k$ , Cartan distribution $C$ is generated by

D_{x^{i}}^{(k)} := \partial x_{i} + \sum_{α = 1}^{q} \sum_{0 \leq | J | < k} u_{J, i}^{α} \partial_{u_{J}^{α}}, 1 \leq i \leq p

V_{α}^{J} := \partial_{u_{J}^{α}}, | J | = k, 1 \leq α \leq q .

Cartan distribution restricted to the submanifold $S$ of $J^{k} (E)$ given by a system of DEs is known as the Vessiot distribution. See also [Vitagliano 2017] page 22, where it defines the distribution:

C (S) : z \mapsto C (S)_{z} := C_{z} \cap T_{z} S

In general, is not Frobenius integrable, but Vessiot gave a method to construct all the integrable subdistributions for a given distribution.

The integral manifolds of this distribution are the solutions of system of DEs (provided they have the proper dimension, I guess...)