Axially symmetric exponential metric

We present a new family of vacuum axially symmetric static solutions to Einstein's field equations. This solution is a geometrically modified version of the Yilmaz exponential metric (YEM) which has recently been shown to represent a traversable wormhole. This vacuum solution is a one-parameter static singular solution with its singularity located at r = 0. Furthermore, it is shown that both the YEM and its geometrically modified version, introduced in this work i.e., the Axially Symmetric Exponential metric (ASEM) belong to a general 2-parameter class of solutions supported by an anisotropic fluid. Depending on the parameters the generic energy-momentum tensor represents either an exotic or a normal fluid. Concerning the ASEM, the circular stable orbit does not exist for a null particle, however, for a massive particle such orbit exists provided its radius is larger than a minimum radius. This is in analogy with the Schwarzschild black hole as well as the YEM.