into the framework. A priori this can be done in two different ways. Either one considers $q$ as a third (a priori small) parameter which “turns on” charge. In this case one still perturbs off a spherically symmetric, static, uncharged solution of the Vlasov-Poisson system. The other way is to use the fact that in the non-relativistic limit the Maxwell equations reduce to the Poisson equation as well and one perturbs around a charged solution of the Vlasov-Poisson system. It turns out that the first approach is easier from a technical point of view since the operator $\mathfrak F$ that the implicit function theorem will be applied to is changed only insignificantly by the included Maxwell equations. However, the result would be restricted to small particle charge parameters $q$. In the second approach arbitrary values $0 \leq q < m_p$ of the particle charge parameter can be treated, where $m_p$ denotes the mass of the particles. In this case the operator $\mathfrak F$ has additional terms. In this article the second approach is presented.

In an axially symmetric, static setting the EVM-system reduces to a system of coupled, non-linear Poisson equations in different dimensions and a first order PDE. The 