Isotopy Classification of Engel Structures on Circle Bundles

We call two Engel structures isotopic if they are homotopic through Engel structures by a homotopy that fixes the characteristic line field. In the present paper we define an isotopy invariant of Engel structures on oriented circle bundles over closed oriented three-manifolds and apply it to give an isotopy classification of Engel structures on circle bundles with characteristic line field tangent to the fibers.