  For regime-switching diffusions processes with singular drifts, we introduce
integrability conditions involving a nice reference probability measure and the
$Q$-matrix of the jump part to study the existence of the invariant probability
measures. Consequently, the generator of the regime-switching diffusions
process has an extension in the $L^1$-space w.r.t the invariant probability
measure to be a generator of $C_0$-semigroup. Moreover, we prove the uniqueness
of the extension. Regularities and the uniqueness of the invariant probability
density w.r.t the nice reference probability measure are also considered.
