The group H of the internal symmetries of the axisymmetric field equations in general relativity is known to be isomorphic to SO(2,1), which is the double covering of the conformal group of the hyperbolic complex plane H. The Ernst potential ξ can then be geometrically understood as a map ξ:R3/SO(2) → H. The fact that the hyperbolic plane is split into two connected components is used to introduce an algebraic invariant n∈Z+ for every axisymmetric solution. It is shown that under reasonable hypotheses this invariant is related to the number of S1 curves where the manifold is intrinsically singular.
