below, we are going to introduce [*source functions*]{} to collect these terms. This allows to present the reduced system in a compact way and also facilitates the presentation of the subsequent analysis. Moreover, we will introduce [*matter functions*]{} which basically consist in combinations of components $T_{\mu\nu}$ of the Vlasov part of the energy momentum tensor, as in [@akr14].

In the subsequent analysis it will be necessary to show different properties of the matter functions and the source functions, like regularity with respect to the coordinates $\varrho$ and $z$, decay properties, symmetries, or Fréchet differentiability with respect to the solution functions $\zeta$. This means that at some occasions the source functions and the matter functions have to be seen as functions of $\varrho$ and $z$ which are parameterised by the solution functions. At other occasions they have to be seen as functions which take both the coordinate $\varrho$ and the solution functions $\zeta$ (and their derivatives) as arguments. Moreover, for the analysis of the matter functions several different integral representations will be necessary. In order to give a clear presentation we deem it favourable to resort to symbolic notation in a larger extent than in [@akr14].

Now we define the matter 