Minimal neighborhood size for total-integrability rigidity near the boundary

Show that the conclusion of Theorem \ref{thm:birkhoff-1/4} remains valid when total integrability is assumed on a smaller neighborhood of the boundary, possibly under higher-order symmetry assumptions.

Background

In Theorem \ref{thm:birkhoff-1/4}, total integrability is required on a finite boundary-adjacent region of phase space, not an arbitrarily small one.

The authors query whether the same rigidity can be proven when the integrable region is smaller, and whether additional symmetry might compensate for the reduced domain.