The nullity distributions of the two curvature tensors \, $\overast{R}$ and $\overast{P}$ of the Chern connection of a Finsler manifold are investigated. The completeness of the nullity foliation associated with the nullity distribution $\N_{R^\ast}$ is proved. Two counterexamples are given: the first shows that $\N_{R^\ast}$ does not coincide with the kernel distribution of \, $\overast{R}$; the second illustrates that $\N_{P^\ast}$ is not completely integrable. We give a simple class of a non-Berwaldian Landsberg spaces with singularities.

Major modifications, An Example added at the end of the paper, Some Maple calculations inserted