  In this paper, we consider a Finsler sphere $(M,F)=(S^n,F)$ with the
dimension $n>1$ and the flag curvature $K\equiv 1$. The action of the connected
isometry group $G=I_o(M,F)$ on $M$, together with the action of $T=S^1$
shifting the parameter $t\in \mathbb{R}/\mathbb{Z}$ of the closed curve $c(t)$,
define an action of $\hat{G}=G\times T$ on the free loop space $\lambda M$ of
$M$. In particular, for each closed geodesic, we have a $\hat{G}$-orbit of
closed geodesics. We assume the Finsler sphere $(M,F)$ described above has only
finite orbits of prime closed geodesics. Our main theorem claims, if the
subgroup $H$ of all isometries preserving each close geodesic has a dimension
$m$, then there exists $m$ geometrically distinct orbits $\mathcal{B}_i$ of
prime closed geodesics, such that for each $i$, the union ${B}_i$ of geodesics
in $\mathcal{B}_i$ is a totally geodesic sub-manifold in $(M,F)$ with a
non-trivial $H_o$-action. This theorem generalizes and slightly refines the one
in a previous work, which only discussed the case of finite prime closed
geodesics. At the end, we show that, assuming certain generic conditions, the
Katok metrics, i.e. the Randers metrics on spheres with $K\equiv 1$, provide
examples with the sharp estimate for our main theorem.
