solution of the reduced EVM-system is determined by the collection $\zeta = (\nu, h, \xi, \omega, A_t, a) \in \mathcal X$ of six functions, defined in a suitably chosen function space $\mathcal X$ (defined in Section \[sect\_function\_space\] below). Proposition \[prop\_equivalent\] below states that a solution of the reduced EVM-system with any parameter $c$ can be converted into an axially symmetric, stationary solution of the EVM-system with the parameter $c=1$.

[*Step 3: Introduction of the solution operator $\mathfrak F$.*]{} A solution of the reduced EVM-system with parameters $\gamma:= c^{-2}, \lambda \in  [0,1) \times (-1,1)$ is then obtained as perturbation of a spherically symmetric solution of the Vlasov-Poisson system. This spherically symmetric solution of the Vlasov-Poisson system we denote by $\zeta_0\in\mathcal X$.

To this end in Section \[sect\_def\_f\] an operator $\mathfrak F: \mathcal X \times [0,1) \times (-1,1) \to \mathcal X$ with the following properties is defined. Firstly, a collection of functions $\zeta\in\mathcal X$ is a solution of the reduced EVM-system with parameters $\gamma$, $\lambda$ if and only if $\mathfrak F[\zeta; \gamma, \lambda] = 0$. (The “if”-direction is essential.) Secondly, $\mathfrak F[\zeta_0; 0, 0] = 0$. In Section \[sect\_well\_defined\] we show that this operator is well defined. The mentioned properties are shown 