Parallel Nilpotent Endomorphisms Without Parallel Null Vectors

We construct a closed complete flat pseudo-Riemannian manifold of dimension 16 and signature (8,8) carrying a nonzero parallel self-adjoint endomorphism N with N² = 0, although neither the manifold nor any connected double cover admits a nonzero parallel vector field. This gives a negative answer, as stated, to Question 12.0.3 in the Burns–Matveev list of open problems about geodesics. The construction starts with a compact flat Riemannian manifold whose holonomy has odd order and zero fixed space, doubles its Euclidean representation, equips the double with the split metric, and sets N(u,v) = (0,u). Odd-order holonomy survives every index-two subgroup. An explicit input is the free (ℤ/3ℤ)² quotient of an Eisenstein four-torus constructed by Bauer and Gleissner. We also show that when the top image of a parallel nilpotent has rank one, the manifold or its orientation double cover does carry a parallel null vector; in particular, the original question is affirmative in Lorentzian and co-Lorentzian signature. Research status: Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-use disclosure: AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text.