${\mathcal{C}}^{\mathrm{\infty }}$-symmetry of distribution

Definition
A nowhere-vanishing vector field $X$ on $M$ is called a ${\mathcal{C}}^{\mathrm{\infty }}$-symmetry of an involutive distribution $\mathcal{Z}=\mathcal{S}(\{Z_1,\dots ,Z_r\})$ if:

1. $\{Z_1,\dots ,Z_r,X\}$ is independent for every $p\in M$,
2. there exist smooth functions ${\lambda }_i,c_i^k$ ($i,k=1,\dots ,r$) such that

Alternative definition
Let $\mathcal{D}$ be a distribution on a manifold $M$. A vector field $X\in \mathfrak{X}(M)$ is called a ${\mathcal{C}}^{\mathrm{\infty }}$-symmetry of $\mathcal{D}$ if

In other words, for any vector field $Y\in \mathcal{D}$, the Lie bracket $[X,Y]$ can be expressed as a linear combination of vector fields in $\mathcal{D}$ and $X$ itself, with coefficients in ${\mathcal{C}}^{\mathrm{\infty }}(M)$.

Remarks
It is a generalization of the idea of symmetry of a distribution, as they consist of distribution symmetries which, although their flow does not behave so well (mapping integral submanifolds to integral submanifolds), it is not too bad. In fact, all ${\mathcal{C}}^{\mathrm{\infty }}$-symmetries of distribution can be corrected, to be converted into a symmetry of a distribution, according to the symmetrizing factor lemma.

Applied to ODE, this concept gives rise to generalized Cinf-symmetry ODE.

When we chain several cinf-symmetry of distribution we obtain a cinf-structure for $\mathcal{Z}$.

Do they have the IBD property for distributions?

It is a key element in the Geometric integration of differential equations.

An equivalent definition entirely in terms of 1-forms, using the dual Pfaffian system ${\mathcal{Z}}^{\circ }$, is given in cinf-symmetry of a Pfaffian system.