  We solve multidimensional SDEs with distributional drift driven by symmetric,
$\alpha$-stable L\'evy processes for $\alpha\in (1,2]$ by studying the
associated (singular) martingale problem and by solving the Kolmogorov backward
equation. We allow for drifts of regularity $(2-2\alpha)/3$, and in particular
we go beyond the by now well understood "Young regime", where the drift must
have better regularity than $(1-\alpha)/2$. The analysis of the Kolmogorov
backward equation in the low regularity regime is based on paracontrolled
distributions. As an application of our results we construct a Brox diffusion
with L\'evy noise.
  Keywords: Singular diffusions, stable L\'evy noise, distributional drift,
paracontrolled distributions, Brox diffusion
