There are proofs that there is only one spherically symetric solution.
So what happens with the putting more mass in one side problem?
Diagrams – kruskal + some others. – attempt join
Equations – attempt join
My conjecture is that when you have a full solution – the solution with asymetric mass is not allowed – the mass will even up eventually, but it will vibrate to equibrium. In reality – the equilbrium never arrives.