Let $\mathcal{E}^3 \subset TM^n$ be a smooth $3$-distribution on a smooth $n$-manifold, and $W \subset \mathcal{E}$ a line field such that $[\mathcal{W}, \mathcal{E}] \subset \mathcal{E}$.We give a condition for the existence of a plane field $\mathcal{D}^2$ such that $\mathcal{W} \subset\mathcal{D}$ and $[\mathcal{D},\mathcal{D}] = \mathcal{E}$ near a closed orbit of $\mathcal{W}$. If $\mathcal{W}$ has a non-singular Morse–Smale section, we get a condition for the global existence of $\mathcal{D}$. As a corollary we obtain conditions for a non-singular vector field $\mathcal{W}$ on a $3$-manifold to be Legendrian, and for an even contact structure $\mathcal{E} \subset TM^4$ to be induced by an Engel structure $\mathcal{D}$.

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关于一些与不可积平面场相切的矢量场的动力学

设 $\mathcal{E}^3 \subset TM^n$ 是光滑 $n$-流形上的光滑 $3$-分布，$W \subset \mathcal{E}$ 是一个线场，使得 $[\ mathcal{W}, \mathcal{E}] \subset \mathcal{E}$。我们给出一个平面场 $\mathcal{D}^2$ 的存在条件，使得 $\mathcal{W} \subset \mathcal{D}$ 和 $[\mathcal{D},\mathcal{D}] = \mathcal{E}$ 靠近 $\mathcal{W}$ 的闭合轨道。如果 $\mathcal{W}$ 有一个非奇异的 Morse–Smale 截面，我们就得到了 $\mathcal{D}$ 全局存在的条件。作为推论，我们获得了在 $3$-流形上的非奇异向量场 $\mathcal{W}$ 是 Legendrian 的条件，并且对于偶数接触结构 $\mathcal{E} \subset TM^4$ 是由恩格尔结构 $\mathcal{D}$ 诱导。