  We study skew products where the base is a hyperbolic automorphism of
$\mathbb{T}^2$, the fiber is a smooth area preserving flow on $\mathbb{T}^2$
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 $K$ and not Bernoulli. As a consequence we get a natural class
of volume-preserving diffeomorphisms of $\mathbb{T}^4$ which are $K$ and not
Bernoulli.
