On the dynamics of some vector fields tangent to non-integrable plane fields

Let $\mathcal{E}^3\subset TM^n$ be a smooth $3$-distribution on manifold of dimension $n$ and $\mathcal{W}\subset\mathcal{E}$ line field such that $[\mathcal{W},\mathcal{E}]\subset\mathcal{E}$. Under some orientability hypothesis, we give necessary condition for the existence plane $\mathcal{D}^2$ $\mathcal{W}\subset\mathcal{D}$ $[\mathcal{D},\mathcal{D}]=\mathcal{E}$. Moreover study case where section $\mathcal{W}$ is non-singular Morse-Smale get sufficient global $\mathcal{D}$. As corollary conditions vector $W$ $3$-manifold to Legendrian contact structure Similarly with these techniques can when an even $\mathcal{E}\subset TM^4$ induced by Engel