  It is proved that all special flows over the rotation by an irrational
$\alpha$ with bounded partial quotients and under $f$ which is piecewise
absolutely continuous with a non-zero sum of jumps are mildly mixing. Such
flows are also shown to enjoy a condition which emulates the Ratner condition
introduced in \cite{Rat}. As a consequence we construct a smooth vector--field
on $\T^2$ with one singularity point such that the corresponding flow
$(\phi_t)_{t\in\R}$ preserves a smooth measure, its set of ergodic components
consists of a family of periodic orbits and one component of positive measure
on which $(\phi_t)_{t\in\R}$ is mildly mixing and is spectrally disjoint from
all mixing flows.
