  Introducing certain singularities, we generalize the class of one-dimensional
stochastic differential equations with so-called generalized drift. Equations
with generalized drift, well-known in the literature, possess a drift that is
described by the semimartingale local time of the unknown process integrated
with respect to a locally finite signed measure \nu. The generalization which
we deal with can be interpreted as allowing more general set functions \nu, for
example signed measures which are only \sigma-finite. However, we use a
different approach to describe the singular drift. For the considered class of
one-dimensional stochastic differential equations, we derive necessary and
sufficient conditions for existence and uniqueness in law of solutions.
