We study skew products where the base is a hyperbolic automorphism of \begin{document}$\mathbb{T}^2$\end{document} , the fiber is a smooth area preserving flow on \begin{document}$\mathbb{T}^2$\end{document} with one fixed point (of high degeneracy) and the skewing function is a smooth non coboundary with non-zero integral. The fiber dynamics can be represented as a special flow over an irrational rotation and a roof function with one power singularity. We show that for a full measure set of rotations the corresponding skew product is \begin{document}$K$\end{document} and not Bernoulli. As a consequence we get a natural class of volume-preserving diffeomorphisms of \begin{document}$\mathbb{T}^4$\end{document} which are \begin{document}$K$\end{document} and not Bernoulli.