  In this paper, as an application of the inverse problem of calculus of
variations, we investigate two compatibility conditions on the spherically
symmetric Finsler metrics. By making use of these conditions, we focus our
attention on the Landsberg spherically symmetric Finsler metrics. We classify
all spherically symmetric manifolds of Landsberg or Berwald types. For the
higher dimensions $n\geq 3$, we prove that: all Landsberg spherically symmetric
manifolds are either Riemannian or their geodesic sprays have a specific
formula; all regular Landsberg spherically symmetric metrics are Riemannian;
all (regular or non-regular) Berwald spherically symmetric metrics are
Riemannian. Moreover, we establish new unicorns, i.e., new explicit examples of
non-regular non-Berwaldian Landsberg metrics are obtained. For the
two-dimensional case, we characterize all Berwald or Landsberg spherically
symmetric surfaces.
