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 \(N^2=0\) , 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 \((\mathbb Z/3)^2\) 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.