I think OP wants to know if there exists a general solution to the two body problem. You wouldn't have very much luck even writing down the field equations, because no assumptions of symmetry or otherwise simplification exist in the two-body problem to bring them to a tractable form. Usually people approach this perturbatively to develop an approximation by iteration (see Post-Newtonian Expansion) or use numerical methods. As someone already mentioned, there are special cases of a two-body problem which admit general solutions, but nobody's found a solution to the general two-body problem. There's really not even a good general solution to the one-body problem, since the only ways we approach it are through symmetric solutions! It's easy to understand this lack of solutions from a bit more of a mathematical perspective. The vacuum field equations simply say that a vacuum region of spacetime containing matter is a generic Ricci flat four-dimensional Lorentzian manifold. There are entire classes of such objects, so it's not as easy as in, say, electromagnetism, to just write down a general solution in terms of Fourier modes.