In this paper, we focus our study on the Berwald curvature G of an n-dimensional Landsberg–Finsler manifold (M, F). We show that the image of the Berwald curvature of F lies in the vertical part of the holonomy distribution \(\mathcal {V}_{{{{\mathcal {D}}}_{{\mathcal {H}}}}}\) associated with the geodesic spray of F whenever the Riemann curvature is non-zero. Furthermore, regardless of the vanishing property of the Riemann curvature, the image of the Berwald curvature \(\operatorname {Im}(G)\) of a Landsberg metric lies in the vertical part \(\mathcal {V}_{\ker dF}\) of the kernel of the Finsler function. We prove that either the geodesic spray of a Landsberg metric is uniquely metrizable by a Landsberg metric or the rank of Berwald curvature is at most \(n-2\) . We discover that the rank of \(\operatorname {Im}(G)\) of two classes of the unicorn’s Landsberg metrics is at most \(n-2\) . We investigate the nullity and kernel distributions of G for the two classes of the unicorn’s Landsberg metrics. Some explicit examples of “singular unicorns”, i.e. singular Landsberg metrics are considered and discussed. This work is ended by some concluding remarks on the unicorn’s problem.
