  Let $\Omega$ be a bounded centrally symmetric domain in $\mathbb{R}^2$ with
analytic boundary $\partial \Omega$ and center $c$. Let $\tau = \tau(\Omega)$
be the number of points $p$ on $\partial \Omega$ such that the normal line to
$\partial \Omega$ at $p$ passes through $c$. We show that if $\tau < 8$ then
$\Omega$ satisfies the Schiffer's conjecture.
