  The zero-noise limit of differential equations with singular coefficients is
investigated for the first time in the case when the noise is an $\alpha
$-stable process. It is proved that extremal solutions are selected and the
respective probability of selection is computed. For this purpose an exit time
problem from the half-line, which is of interest in its own right, is
formulated and studied by means of a suitable decomposition in small and large
jumps adapted to the singular drift.
