  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 $\gamma$.
