  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 $\partial D$, assume that the
symplectic billiard map has a (simple) continuous invariant curve $\delta
\subset \mathcal{P}$ of rotation number $1/4$ (winding once around $\partial
D$) 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 $\partial D$ 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.
