Extraordinary vacuum black string solutions

In addition to the boosted static solution there are two other classes of stationary string-like solutions of the vacuum Einstein equation in (4+1)-dimensions. Each class is characterized by three parameters of mass, tension, and momentum flow along the fifth coordinate. We analyze the metric properties of one of the two classes, which was previously assumed to be naked singular, and show that the solution spectrum contains black string and wormhole in addition to the known naked singularity as the momentum flow to mass ratio increases. Interestingly, there does not exist new zero momentum solution in these cases.