Engel CR submanifolds of ${\mathbb{C}}^3$

We give a sufficient condition for an ${\mathbb{S}}^1$-bundle over a $3$-manifold to admit an immersion (or embedding) into ${\mathbb{C}}^3$ so that its complex tangencies define an Engel structure. In particular, every oriented ${\mathbb{S}}^1$-bundle over a closed, oriented $3$-manifold admits such an immersion. If the bundle is trivial, this immersion can be chosen to be an embedding and, moreover, infinitely many pairwise smoothly non-isotopic embeddings of this type can be constructed. These are the first examples of compact submanifolds of ${\mathbb{C}}^3$ whose complex tangencies are Engel, answering a question of Y. Eliashberg.