We prove there is a class of maps $\gamma:\mathbb{ T}^{ 2n}\rightarrow\mathbb{ S}^1$ 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 $\gamma$.
Categories: math.DS