On the Integrability of codimension-one invariant subbundles of partially hyperbolic skew-products

We prove there is a class of maps $\gamma :{\mathbb{T}}^{2n}\to {\mathbb{S}}^1$ such that a conservative dynamically coherent partially hyperbolic skew-product on ${\mathbb{T}}^{2n}\times {\mathbb{S}}^1$ with fixed hyperbolic dynamics on the base and rotation by angle $\gamma$ acting on the fibers have integrable hyperbolic structure which also implies in particular that they are not contact diffeomorphisms. In dimension $3$, we prove the same result using a standard technique in Contact Geometry, namely, that of \emph{characteristic foliations}, which gives a simple proof of the result but with more tight restrcitions to the map $\gamma$.