I am looking for references that can (or an expert who is willing to) suggest methods for finding stationary, axisymmetric solutions to a set of nonlinear field equations related to, but not quite the same as the Einstein-Maxwell equations. It's easy enough to write down the time-independent form of the equations, but I'm not sure where to go from there. The equations are:
$$ \partial_\mu \partial_\mu A^\nu= j^\nu, $$ where
$$ j^\nu=K(F_{\alpha\beta} F^{\beta\nu} -1/4 (F_{\mu\beta} F^{\mu\beta}) \delta^{\alpha\nu} )\epsilon_\alpha, $$
$$ F_{\alpha\beta} = \partial_\mu A_\alpha -\partial_\alpha A_\beta, $$
and $\epsilon_\alpha$ is the timelike unit eigenvector of $F_{\mu\beta}.$
The constant K is an adjustable parameter. The field variables are complex vectors $A_\alpha$, and the products are as shown, not products of complex conjugates. "Stationary solution" in this context means any $A_\alpha (x_\gamma) $ that is a solution to the equations and is either independent of $x_o$ or is periodic in $x_o$. The solutions also need to be stable.