the electro-magnetic field.*]{} The assertion that the solution comprises a poloidal magnetic field if and only if the solution is rotating follows from the structure of the reduced EVM-system, see Remark \[rem\_mag\_rot\]. For the assertion that there is no toroidal magnetic field, see Lemma \[lem\_no\_tor\].

Axial symmetry {#sect_axial}
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Let $x^i$, $i=1,\dots,n$ be coordinates on $\mathbb R^n$. A function $f:\mathbb R^n \to \mathbb R$ is axially symmetric around the $x^n$-axis if and only if there exists a function $\hat f: [0,\infty) \times \mathbb R \to \mathbb R$ such that $$f\left(x^1,\dots,x^n\right) = \hat f\left(\varrho(x^1, \dots, x^{n-1}), x^n\right),$$ where $$\label{def_rho}
\varrho(x^1, \dots, x^{n-1}) := \sqrt{\left(x^1\right)^2 + \dots + \left(x^{n-1}\right)^2}.$$ By abuse of notation, we will use the same symbol for the original function on $\mathbb R^2$, $\hat f$ in this example, and the induced axially symmetric functions $f$ on $\mathbb R^n$ for different dimensions $n$.

\[rem\_even\] At some places in the analysis presented in this article it will be useful to view an axially symmetric function $f:\mathbb R^n \to \mathbb R$ as a function in $\varrho$ and $z$ defined on $\mathbb R^2$, by extending it as even function to negative values of $\varrho$. The obtained function on $\mathbb R^2$ then has the 