The monodromy matrix method of solving an exterior boundary value   problem for a given stationary axisymmetric perfect fluid solution
  A procedure is described for matching a given stationary axisymmetric perfect fluid solution to a not necessarily asymptotically flat vacuum exterior. Using data on the zero pressure surface, the procedure yields the Ernst potential of the matching vacuum metric on the symmetry axis. From this the full metric can be constructed using a variety of well established procedures. 
I. Hauser and F. J. Ernst