It is possible to get the Schwartzschild metric assuming spherical symmetry, vacuum solution and Minkowski spacetime when $r \to \infty$.
Is it possible an analytic solution for a geocentric system? I mean, taking the apparent daily movement of the celestial bodies as real. So, the (apparent) trajectories of moon, sun and planets should be geodesics according to the metric.
I suppose it is necessary to assume that when $r \to \infty$ geodesics are circles, as the fixed stars does every night from an observer on earth. So it is not a Minkowski metric at infinity.
I don't know if the Godel solution is something like that.