  In this paper, we prove that for every non-degenerate $C^3$ compact
star-shaped hypersurface $\Sigma$ in $\mathbf{R}^{6}$ which carries no prime
closed characteristic of Maslov-type index $0$ or no prime closed
characteristic of Maslov-type index $-1$, there exist at least three prime
closed characteristics on $\Sigma$.
