  When a closed Finsler manifold admits continuous isometric actions,
estimating the number of orbits of prime closed geodesics seems a more
reasonable substitution for estimating the number of prime closed geodesics. To
generalize the works of H. Duan, Y. Long, H.B. Rademacher, W. Wang and others
on the existence of two prime closed geodesics to the equivariant situation, we
purpose the question if a closed Finsler manifold has only one orbit of prime
closed geodesic if and only if it is a compact rank-one Riemannian symmetric
space. In this paper, we study this problem in homogeneous Finsler geometry,
and get a positive answer when the dimension is even or the metric is
reversible. We guess the rank inequality and algebraic techniques in this paper
may continue to play an important role for discussing our question in the
non-homogeneous situation.
