Bialy-Mironov type rigidity for centrally symmetric symplectic billiards

The aim of the present paper is to establish a Bialy-Mironov type rigidity for centrally symmetric symplectic billiards. For a centrally symmetric $C^2$ strongly-convex domain $D$ with boundary $\mathrm{\partial }D$, assume that the symplectic billiard map has a (simple) continuous invariant curve $\delta \subset \mathcal{P}$ of rotation number $1\mathrm{/}4$ (winding once around $\mathrm{\partial }D$) and consisting only of $4$-periodic orbits. If one of the parts between $\delta$ and each boundary of the phase-space is entirely foliated by continuous invariant closed (not null-homotopic) curves, then $\mathrm{\partial }D$ is an ellipse. The differences with Birkhoff billiards are essentially two: it is possible to assume the existence of the foliation in one of the parts of the phase-space detected by the curve $\delta$, and the result is obtained by tracing back the problem directly to the totally integrable case.