and shorthands shall be introduced. Partial derivatives $\partial_\varrho \nu$, $\partial_z A_\varphi$, etc. will be denoted as $\nu_{,\varrho}$, $A_{\varphi,z}$, etc. We define the functions $\xi, h, a$ by the following changes of variables: $$\begin{aligned}
\xi &= \mu + \gamma \nu, \\
H &= 1 + h, \\
A_\varphi &= \varrho^2 a.\end{aligned}$$ Further, we call $(\nu, h, \xi, \omega, A_t, a)$ the [*solution functions*]{} and in the remainder of this article we will use the shorthand $$\zeta := (\nu, h, \xi, \omega, A_t, a).$$ We do not include the components $A_\varrho$ and $A_z$ of the four-potential $A$ into the solution functions $\zeta$ since it will turn out that in the current setting they must vanish everywhere, cf. Lemma \[lem\_no\_tor\] below.

In [@akr11; @akr14], where the existence of axially symmetric solutions of the Einstein-Vlasov system with uncharged particles is proven, a reduced system of equations is considered as well. The reduced EVM-system presented below coincides with the reduced system in [@akr14] if the charge parameter $q$ is set to zero. When the Maxwell equations are added to the framework not only the number of equations increases but also the number of terms in the Einstein equations increases by a multiple. For this reason, 