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

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