  So far, it is still unknown whether all the closed characteristics on a
symmetric compact star-shaped hypersurface $\Sigma$ in ${\bf R}^{2n}$ are
symmetric. In order to understand behaviors of such orbits, in this paper we
establish first two new resonance identities for symmetric closed
characteristics on symmetric compact star-shaped hypersurface $\Sigma$ in ${\bf
R}^{2n}$ when there exist only finitely many geometrically distinct symmetric
closed characteristics on $\Sigma$, which extend the identity established by
Liu and Long in \cite{LLo1} of 2013 for symmetric strictly convex
hypersurfaces. Then as an application of these identities and the identities
established by Liu, Long and Wang recently in \cite{LLW1} for all closed
characteristics on the same hypersurface, we prove that if there exist exactly
two geometrically distinct closed characteristics on a symmetric compact
star-shaped hypersuface in ${\bf R}^4$, then both of them must be elliptic.
