Vector field determining a family of codimension 1 submanifolds perpendicular to the field

Question

Given a smooth vector field in a compact domain contained in $\mathbb{R}^{n}$, under what conditions can we conclude that this determines uniquely a family of $(n-1)$-dimensional smooth submanifolds so that at each point on each submanifold, the unit normal is in the direction of the vector field value at that point?

Answer 1

Turn the vector field $V$ into a $1$-form $\omega$ in the obvious way. Then you will get integrability iff $d\omega\wedge\omega=0$.

(In the case $n=3$ you could state this directly as $\text{curl}\,V\times V=0$.)