The Kruskal-Szekeres solution extends the exterior Schwarzschild solution maximally, so that every geodesic not contacting a curvature singularity can be extended arbitrarily far in either direction.
Wikipedia says it is the unique maximal extension that is an analytic, simply connected vacuum solution.
So what happens without "simply"? I.e. anyone know a maximal extension that is an analytic, connected vacuum solution, but is not simply connected?