formulated differently due to the additional Maxwell equations. We comment on the modifications in the respective sections.

[*Step 1: Elimination of the Vlasov equation.*]{} For the particle distribution function we use the ansatz $f(x,p) = \phi(E(x,p))\psi(\lambda, L(x,p))$, see (\[ansatz\_f\]) below. So the particle distribution depends only on the particle energy $E(x,p)$ and the $z$-component of the angular momentum $L(x,p)$, see the definitions (\[formula\_e\]) and (\[formula\_l\]) below. Since the quantities $E$ and $L$ are conserved along its characteristics the Vlasov equation is automatically satisfied for such an ansatz, cf. Section \[sect\_char\] below. Furthermore, we introduce a parameter $\lambda$ which “turns on” the dependency of $f$ on $L$. This means that if $\lambda = 0$ then $\psi \equiv 1$, i.e. for each value of the $z$-component of the angular momentum there are equally many particles.

[*Step 2: Reduction of the remaining system.*]{} First we express the EVM-system (\[eq\_einstein\])–(\[maxwell\_eq\_2\]) in cylindrical coordinates. The assumptions that the solution is asymptotically flat, axially symmetric, and time independent yield simplifications of the system of equations. We call this simplified system the [*reduced EVM-system*]{}, cf. Definition \[def\_red\_evm\] below, and it is stated in Section \[sect\_red\_sys\], equations (\[final\_eq\_nu\])–(\[bc\_center\]), below, where any value of $c\in (0,\infty)$ is admitted. The 