solution of this system consists in a collection of functions which we denote $\zeta$. For the construction of a well defined solution operator $\mathfrak F$ one has to assure for that the source terms of these Poisson equations are sufficiently regular. However, after the variable substitution $A_\varphi = \varrho^2 a$ one obtains for the $\varphi$-component of the electro-magnetic four potential $A$ the equation $$\label{intro_poisson_a}
\Delta_5 a = \frac{2}{1+h} \frac{a \partial_\varrho h}{\varrho} + \frac{2}{4\pi^2 c^2} \frac{a \partial_\varrho \nu}{\varrho} + \dots.$$ On the right hand side only some a priori problematic terms are written out explicitly. The functions $\nu$ and $h$ are part of the collection $\zeta$ of solution functions of the EVM-system. These terms are a priori problematic because they are singular at the axis $\varrho = 0$.

Looking a bit closer one notices that the right member of equation (\[intro\_poisson\_a\]) is not singular if $h$ and $\nu$ are axially symmetric functions of a certain regularity. However, by dividing by $\varrho$ one “looses derivatives”. For this reason the function space $\mathcal X$ has to be chosen such that the individual functions of the collection $\zeta$ have a hierarchy in regularity. For equation (\[intro\_poisson\_a\]) for example one needs that $h$ and $\nu$ 