  We establish the splitting lemmas (or generalized Morse lemmas) for the
energy functionals of Finsler metrics on the natural Hilbert manifolds of
$H^1$-curves around a critical point or a critical $\R^1$ orbit of a Finsler
isometry invariant closed geodesic. They are the desired generalization on
Finsler manifolds of the corresponding Gromoll-Meyer's splitting lemmas on
Riemannian manifolds (\cite{GM1, GM2}). As an application we extend to Finsler
manifolds a result by Grove and Tanaka \cite{GroTa78, Tan82} about the
existence of infinitely many, geometrically distinct, isometry invariant closed
geodesics on a closed Riemannian manifold.
