In this paper, we consider a Finsler sphere
$\left(M,F\right)=\left({S}^{n},F\right)$ with dimension
$n>1$ and flag curvature
$K\equiv 1$. The action of the
connected isometry group
$G={I}_{o}\left(M,F\right)$
on
$M$, together with
the action of
$T={S}^{1}$ shifting
the parameter
$t\in \mathbb{R}\u2215\mathbb{Z}$ of the
closed curve
$c\left(t\right)$, define
an action of
$\u011c=G\times T$ on the
free loop space
$\Lambda M$
of
$M$.
In particular, for each closed geodesic, we have a
$\u011c$orbit
of closed geodesics. We assume the Finsler sphere
$\left(M,F\right)$ described
above has only finite orbits of prime closed geodesics. Our main theorem claims that, if the
subgroup
$H$
of all isometries preserving each close geodesic is of dimension
$m$, then there exists
$m$ geometrically distinct orbits
${\mathcal{\mathcal{B}}}_{i}$ of prime closed geodesics,
such that for each
$i$, the
union
${B}_{i}$ of geodesics in
${\mathcal{\mathcal{B}}}_{i}$ is a totally geodesic
submanifold in
$\left(M,F\right)$ with
a nontrivial
${H}_{o}$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.
