Basarab-Horwath 1991 proposition 2

See @basarab
Let
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The form $β_{i}$ is closed if and only if the system ${A, X_{1}, \dots, {\hat{X}}_{i}, \dots, X_{p}}$ is closed and $X_{i}$ is a symmetry of it

Adaptación a mi contexto

Given a codimension 1 involutive distribution $Z = S (Z_{1}, \dots, Z_{n - 1})$ on a $n$ -dimensional manifold $M$ and a vector field $X$ symmetry of $Z$ , the 1-form

ω = \frac{Z_{n - 1} ⌟ \dots ⌟ Z_{1} ⌟ Ω}{X ⌟ Z_{n - 1} ⌟ \dots Z_{1} ⌟ Ω}

where $Ω$ is a volume form and $⌟$ denote contraction, is closed.