  In this article, we derive the explicit transition density functions of skew
Brownian motion (SBM in abbreviation) with two-valued drift for all $t>0$. As
an important step of this result, it is also shown in this paper that SBM with
two-valued drift is a strong Markov process by finding its symmetrizing measure
and canonical scale function, from which one can tell what values of the drift
make such a process transient or recurrent.
