Meromorphic Painlev\'e III transcendents and the Joukowski correspondence

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.