The solutions of the equations of General Relativity are hard, and it is not surprising that only special cases have been analysed in detail. It is more surprising however that the professional cosmologists have chosed to treat an entire Galaxy with just the Newtonian mechanics, and when the results didn’t conform to the measurements, they went ahead, and formulated an entire series of theories with exotic characteristics. New results published recently of a reformulation of Galactic models that take into account the self-effects that General Relativity implies, match observations without the need of dark matter, or other kinds of exoterica.
Mathematics is fertile, but reality is as well: I am sure that all the creativity that went into the models that now seem unnecessary is going to help in other now apparently unrelated fields.