Rigidity from partial total integrability for outer billiards

Establish a rigidity result for outer billiards under the assumption that total integrability holds only on a proper subset of the exterior phase space.

Background

Total integrability on the full exterior phase space implies the boundary is an ellipse, by Theorem \ref{thm:total}.

The authors ask whether a similar conclusion can be reached under weaker, localized total integrability assumptions on only part of the phase space.