Symmetrizing factor in the case of corank 1

(xournal 195)
Key idea to start reasoning:
Lemma
Given a closed 1-form $\omega$ on an open set $U\subset {\mathbb{R}}^n$, a vector field $V$ is a symmetry of the Pfaffian system $\mathcal{S}(\{\omega \})$ if and only if $V⌟\omega$ is a first integral of $\mathcal{S}(\{\omega \})$.
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Proof
The function $V⌟\omega$ is a first integral if and only if $d(V⌟\omega )\wedge \omega =0$. Taking into account Cartan formula this is equivalent to

And this is true if and only if ${\mathcal{L}}_V\omega =\beta \omega$ for some function $\beta$, i.e., $V$ is a symmetry of the Pfaffian system
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From here we can conclude several interesting facts, like those on @sherring1992geometric:

Proposition 1
Given a Frobenius integrable 1-form $\omega$ and a vector field $X\notin \mathcal{S}(\{\omega \}{)}^{\perp }$ then a function $f$ satisfies $fX$ is a symmetry of $\mathcal{S}(\{\omega \})$ (is a symmetrizing factor) if and only if $\mu ={\displaystyle \frac{1}{fX⌟\omega }}$ is an integrating factor of $\omega$.
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Proof
We know that there exist an integrating factor $\tilde{\mu }$, that is, a function such that $\tilde{\mu }\omega$ is closed. Taking into account that $\mathcal{S}(\{\tilde{\mu }\omega \})$=$\mathcal{S}(\{\omega \})$, and according to the Lemma above, $fX$ is a symmetry of $\mathcal{S}(\{\omega \})$ if and only if $fX⌟\tilde{\mu }\omega$ is a first integral. Now observe that

and this is a first integral if and only if ${\displaystyle \frac{1}{fX⌟\omega }}$ is an integrating factor.
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Proposition 2
Given a symmetry $V$, then $1/V⌟\omega$ is an integrating factor.
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Proof
Take $f=1$ in Proposition 1.
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Proposition 3
Given a Frobenius integrable 1-form, a smooth function $\mu$ is an integrating factor if there exists a symmetry $V$ such that $\mu =1/V⌟\omega$.
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Proof
Take any $X$ such that $X⌟\omega \ne 0$. Since $\mu$ is an integrating factor, by Proposition 1 $f={\displaystyle \frac{1}{\mu X⌟\omega }}$ is a symmetrizing factor for $X$, so $V=fX$ is a symmetry. It is easy to check that

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