Classification of the static vacuum metric with Ricci-flat compactification

We explicitly construct all spherically symmetric vacuum solutions of the higher-dimensional Einstein equation in which the size of Ricci-flat compact internal manifolds varies with three-dimensional distance, subject to an asymptotic condition, flat $M_4$ times a compact manifold. Besides the usual Schwarzschild and the trivial vacuum solutions, a variety of new solutions are found, all of which contain a curvature singularity not hidden by an event horizon (naked singularity) unless the extra compact space admits an isometry.