Weyl and papapetrou axisymmetric space-times as geodesics of the hyperbolic complex plane

The problem of axisymmetrie solutions of the Einstein equations depending on a single real function is re-examined. The solutions are geometrically represented as geodesics of the hyperbolic plane. Here intuitive geometry can be applied to show that the gravitational fields found by Papapetrou are generalizations of the Weyl ones, under the action of the isometry group of the hyperbolic plane itself.