It is well-known that Birkhoff's theorem and the classification of LTB spacetimes proves one version of Newton's shell theorem in the context of GR. Another statement in Newtonian gravity, often understood as a shell theorem(*), stipulates that a point mass $m_1$ will attract a homogeneous massive spherical shell $S_2$ as though all the mass of that shell were concentrated at the center of the shell (in the case where $m_1$ is not inside of $S_2$. If that point-mass does happen to be inside of $S_2$, it exerts no net force on the shell). Contrary to the first shell theorem, it is difficult to even formulate a version of the second shell-theorem in GR(**).
With these two shell-theorems, the Newtonian dynamics of the solar system is reduced to a system of interacting point-masses (provided the internal mass-distribution of the planets is almost radially symmetric).
My question: In Newtonian gravity, there is a generalization of the first shell theorem to the shells of a spheroidal shape (Laplace-Ivory theorem) and even more general algebraic surfaces -i.e. the gravitational force vanishes everywhere inside these shells-. Is something similar true and known in GR?
(*) I believe it was also derived by Newton.
(**) The Swiss-cheese cosmologies are testament to a version of the second shell-theorem.
