Cauchy characteristic vector fields

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

Proposition. Cauchy characteristic vector fields constitute a Lie subalgebra of $X (M)$ .
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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 $X ⌟ ω \in I$ for a set of forms $ω$ that generate $I$ 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 $α : R \to M$ is an integral curve of a CCVF $X$ , and $ω \in I^{1}$ , i.e., $ω$ is a 1-form in $I$ . 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 $ω$ with $k \geq 2$ it holds trivially that $α^{*} (ω) = 0$ . 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 $I$ :

A (I)_{x} = {ξ_{x} \in T_{x} M : ξ_{x} ⌟ I_{x} \subset I_{x}}

and

C (I)_{x} = A (I)_{x}^{⊥} \subset T^{*} M =

= {ω \in Λ^{1} (M)_{x} : X ⌟ ω = 0 for all X \in A (I)}

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

I think that $A (I)$ is a generalization to exterior differential system of the associated distribution to a completely integrable Pfaffian system.
Observe that given a 1-form $ω \in I$ , for every $ξ \in A (I)$ we have $ξ ⌟ ω \in I$ , so $ξ ⌟ ω = 0$ 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 $ω \in C (I)$ and

I \cap Ω^{1} \subset C (I)

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

Related ? method of characteristics