  In this paper we investigate the regularity properties of strong solutions to
SDEs driven by L\'evy processes with irregular drift coefficients. Under some
mild conditions, we show that the singular SDE has a unique strong solution for
each starting point and the family of all the solutions forms a stochastic
flow. Moreover, the Malliavin differentiability of the strong solutions is also
obtained. As an application, we also prove a path-by-path uniqueness result for
some related random ODEs.
