Symmetric closed characteristics on symmetric compact convex hypersurfaces in ${\mathbf{R}}^8$: Special cases

Let $\mathrm{\Sigma }$ be a $C^3$ compact symmetric convex hypersurface in ${\mathbf{R}}^8$. For some special cases, we prove that when $\mathrm{\Sigma }$ carries exactly four geometrically distinct closed characteristics, then all of them must be symmetric.