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.