We use the fact that the Papapetron-Majumdar solution of Einstein's equation is static to compute a first integral of motion and use a conformal transformation to convert the remaining equations to the equations of motion of a particle in a potential in flat space, analogously to Maupertuis' principle1).
A practical application of this technique is then provided by means of perturbation theory.
1.) see for example: V.I. Arnol'd: Mathematical Methods of Classical Mechanics