Symmetrizing factor lemma

It could also be concluded from the existence of solutions for a system of first order linear inhomogeneous PDEs.

Theorem. Let $\mathcal{Z}=\mathcal{S}(\{Z_1,\dots ,Z_r\})$ be an involutive distribution on a manifold $M$ of dimension $n$, and let $X$ be a ${\mathcal{C}}^{\mathrm{\infty }}$-symmetry for $\mathcal{Z}$. Then, for every $p\in M$, there exists an open neighbourhood $U$ and a function $f\in {\mathcal{C}}^{\mathrm{\infty }}(U)$ (which we call symmetrizing factor) such that $fX$ is a symmetry for $D$ restricted to $U$.
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Lo que viene a decir es que puedo arreglar el campo $X$ cambiando la longitud del vector (en cada punto) una proporción $f$ de manera que ya su flujo sí lleve integral submanifolds en integral submanifolds.

There is a duality between symmetrising factor and integrating factors module $\mathcal{I}({\mathrm{\Lambda }}_{i_0})$ (it is in the preprint "symmetrizing and integrating factors"). But the search of integrating factors is easier because we are restricted to submanifolds.

And probably, if we know the integrating factors of the restricted 1-forms we can obtain the symmetrising factors and therefore the symmetries of the distribution (see about the substitution of the constants).

The corank 1 case is explained in symmetrizing factors in the case of corank 1.