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 ≥ 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.