  We prove that, under some generic non-degeneracy assumptions, real analytic,
centrally symmetric plane domains are determined by their Dirichlet (resp.
Neumann) spectra. We prove that the conditions are open-dense for real analytic
convex domains. The proof is parallel to the proof that up/down symmetric
domains are spectrally determined. One step is to use a Maslov index
calculation to show that the second derivative of the defining function of a
centrally symmetric domain at the endpoints of a bouncing ball orbit is a
spectral invariant. This is also true for up/down symmetric domains, removing
an assumption from the proof in that case.
