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