Basarab-Horwath 1991 proposition 2

See @basarab
(adapted to my context)

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 $Z$ , then the 1-form

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

where $Ω$ is any volume form and $⌟$ denotes contraction, is closed, and conversely.

See also @lychagin2007contact Example 1.7.1