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In this article, we study complete pseudo-Riemannian manifolds whose coneadmits a parallel symmetric 2-tensorfield. The situation splits in three cases:nilpotent, decomposable or complex Riemannian. In the complex Riemannian anddecomposable cases we provide a classification. In the nilpotent case, we areable to describe completely only a dense open subset of the manifold. Toconclude, we give examples with non-constant curvature in the nilpotent case.

On parallel and symmetric 2-tensorfields on cones over pseudo-Riemannian manifolds