On a conjugacy problem in billiard dynamics

We study symmetric billiard tables for which the billiard map is locally (near an elliptic periodic orbit of period $2$) conjugate to a rigid rotation. In the previous paper (Physica D 255, 31–34 (2013)), we obtained an equation (called below the conjugacy equation) for such tables and proved that if $\alpha $, the rotation angle, is rationally incommensurable with $\pi $, then the conjugacy equation has a solution in the category of formal series. In the same paper there is also numerical evidence that for “good” rotation angles the series have positive radii of convergence. In the present paper we carry out a further study (both analytic and numerical) of the conjugacy equation. We discuss its symmetries, dependence of the convergence radius on $\alpha $, and other aspects.