are of higher regularity than $a$.

This article is a generalisation of [@akr14] and the proof follows the same scheme. Including charge into the framework does not only increase the number of equations in the system but it also increases significantly the number of terms in each equation. Some of these terms require some care in the analysis but clearly not all of them. Still all required properties of the system have to be checked term by term. In order to make the presentation more concise this article resorts more to shorthands and schematic or symbolic notation than [@akr11; @akr14].

In the next section the EVM-system will be introduced. Then, in Section \[sect\_result\], the result of this article will be stated and an outline of the proof will be given. The rest of the article is devoted to the introduction of the technical setup, the definition of the relevant objects, i.e. function spaces and solution operators, and the proofs of important properties of these operators.

The Einstein-Vlasov-Maxwell system
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A solution of the Einstein-Vlasov-Maxwell system (EVM-system) for particles with mass $m_p\geq 0$ and charge $0 \leq q < 1$ is a Lorentzian metric $g\in T^*\mathscr M \otimes T^*\mathscr M$ defined 