Towards some generalization of Birkhoff's theorem

As is known, Birkhoff's theorem states the uniqueness of the solution of Einstein vacuum field equations for an spherical and isolated compact body. Moreover, that solution, the Schwarzschild metric, only possesses one multipole moment which is the mass, and so, Birkhoff's theorem allows to establish a relationship between the spherical symmetry and the multipole structure of the so-called Monopole solution. Hence, the question is whether we can extrapolate this feature of Birkhoff's theorem to any other solution of the Weyl family of axisymmetric vacuum solutions of Einstein field equations. Is there any kind of symmetry which let us characterize the solutions possessing any finite set of multipole moments?.