I will discuss a new approach to solve the Einstein equations in higher dimensions. It consists in decomposing the equations into several sectors with known integrable structure. I will construct the first classes of non-supersymmetric topological solitons that are asymptotic to four-dimensional Minkowski spacetime with additional compact circles. They are induced by bubbles of spacetime supported by electromagnetic flux and look like Reissner-Nordström black holes but are smooth and horizonless. I will pay particular attention to the neutral solutions and compare their geometries to the Schwarzschild black holes. I will also briefly discuss the stability of the solutions.
