  We study a twistor correspondence based on the Joukowski map reduced from one
for stationary-axisymmetric self-dual Yang-Mills and adapt it to the Painlev\'e
III equation. A natural condition on the geometry (axissimplicity) leads to
solutions that are meromorphic at the fixed singularity at the origin. We show
that it also implies a quantisation condition for the parameter in the
equation. From the point of view of generalized monodromy data, the condition
is equivalent to triviality of the Stokes matrices and half-integral exponents
of formal monodromy. We obtain canonically defined representations in terms of
a Birkhoff factorization whose entries are related to the data at the origin
and the Painlev\'e constants.
