I'm looking for ways to find the stable stationary solutions of a modified form of the Einstein-Maxwell equations. "Stationary" would mean that the solution is time-independent; "stable" would mean that if an arbitrary small perturbation were added to the solution its time evolution would return it to the stationary solution. In the standard form of the Einstein-Maxwell equations, the energy-momentum density of the electromagnetic field acts as a source for the gravitational field, but the sources of the electromagnetic field are specified at the outset. In the modified form I'm considering, coupling between the EM and gravitational fields produces an effective charge/current distribution as well as an effective source for the gravitational field; and the fields are otherwise source-free. I expect that any stationary solutions will be axially symmetric. Tips from anyone who has experience with stationary solutions of the standard Einstein-Maxwell equations would be very useful.