  We describe the local conformal geometry of a Lorentzian spin manifold
$(M,g)$ admitting a twistor spinor $\phi$ with zero. Moreover, we describe the
shape of the zero set of $\phi$. If $\phi$ has isolated zeros then the metric
$g$ is locally conformally equivalent to a static monopole. In the other case
the zero set consists of null geodesic(s) and $g$ is locally conformally
equivalent to a Brinkmann metric. Our arguments utilise tractor calculus in an
essential way. The Dirac current of $\phi$, which is a conformal Killing vector
field, plays an important role for our discussion as well.
