Cauchy characteristic vector fields

Definition. Given an exterior differential system $\mathcal{I}$, a Cauchy characteristic vector field (CCVF) is a vector field $X$ such that $X⌟\omega \in \mathcal{I}$ for every $\omega \in \mathcal{I}$.$◼$

Proposition. Cauchy characteristic vector fields constitute a Lie subalgebra of $\mathfrak{X}(M)$ .
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Proof. It can be shown using formula 4 in formulas for Lie derivative, exterior derivatives, bracket, interior product.
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Remarks

• If $X$ is a Cauchy characteristic vector field of an EDS, then it is also a symmetry of the EDS. Therefore, in the case of a Pfaffian system, a Cauchy characteristic vector field is a trivial symmetry of the associated distribution.
• To check that a vector field is a Cauchy characteristic vector fields we only have to check $X⌟\omega \in \mathcal{I}$ for a set of forms $\omega$ that generate $\mathcal{I}$ algebraically.
• The flow lines of a Cauchy characteristic vector field are called the Cauchy characteristics curves of the EDS. It has to do with method of characteristics... Suppose $\alpha :\mathbb{R}\to M$ is an integral curve of a CCVF $X$, and $\omega \in {\mathcal{I}}^1$, i.e., $\omega$ is a 1-form in $\mathcal{I}$. Given ${\partial }_t$, $$\partial_t\lrcorner \alpha^*(\omega)=\alpha'(t)\lrcorner \omega=\mu X\lrcorner \omega=0.$$ On the other hand, for any $k$-form $\omega$ with $k\ge 2$ it holds trivially that ${\alpha }^{\ast }(\omega )=0$. Therefore, the CCVF are those vector fields whose flow lines are the 1-dimensional integral manifolds of the EDS.

In @bryant2013exterior and page 30 and Barco thesis page 33 it is defined at a point $x\in M$ the Cauchy characteristic space of an ideal $\mathcal{I}$:

and

This is called the retracting space at $x$, or the Cartan system of $\mathcal{I}$ (Barco, M. A. thesis, page 34).
The differential system defined by $C(\mathcal{I})$ or, what is the same, the distribution $A(\mathcal{I})$ is completely integrable (see @bryant2013exterior Proposition 2.1. page 31, although it can be concluded from Proposition above).

I think that $A(\mathcal{I})$ is a generalization to exterior differential system of the associated distribution to a completely integrable Pfaffian system.
Observe that given a 1-form $\omega \in \mathcal{I}$, for every $\xi \in A(\mathcal{I})$ we have $\xi ⌟\omega \in \mathcal{I}$, so $\xi ⌟\omega =0$ since the contraction is a 0-form, and they are usually not considered in exterior differential systems (if they are present, we can restrict to the manifold defined by its zeros). Therefore $\omega \in C(\mathcal{I})$ and

A Cauchy characteristic is an integral manifold of the retracting space. They are a special kind of integral manifold of the original EDS.