Linear perturbations of a Schwarzschild black hole by thin disc

The article written by C.M.Will (1974) describes a method how to obtain a perturbation of an (originally) Schwarzschild metric by a slowly rotating light ring. This scheme is revisited in order to adapt it to the case of a thin disc.It turns out for some class of mass distribution on the disc bounded between some radii the solution (or at least its decomposition into spherical harmonics) can be found explicitly.