due to the electro-magnetic field. We have $$E = \tilde E - \frac{1}{\gamma} + qA_t, \qquad L = \tilde L - qA_\varphi.$$

\[cor\_char\] Every function $f:\mathscr P \to \mathbb R_+$ which can be expressed as $$\label{product_strucutre}
f(t,\varrho,\varphi,z,v^0,v^1,v^2,v^3) = \phi(E) \tilde \psi(L)$$ with some functions $\phi, \tilde \psi \in C^1(\mathbb R ; \mathbb R_+)$, solves the Vlasov equation (\[eq\_vlasov\]) and is axially symmetric and time independent.

Since $\mathfrak T E = \mathfrak T L = 0$ we have by the chain rule $\mathfrak Tf = 0$. The remaining asserted properties of $f$ are inherited from the metric functions $\nu$, $\mu$, $H$, and $\omega$.

A more general statement than Corollary \[cor\_char\] is true, for ansatz functions that do not have the product structure (\[product\_strucutre\]). The corollary is however stated this way because in this article only ansatz functions of the form (\[product\_strucutre\]) are considered.

From now on we work with the ansatz $$\label{ansatz_f}
f(x,v) = \phi\left(E \right)\psi(\lambda, L),$$ where $E$ and $L$ are the conserved quantities, given in (\[formula\_e\]) and (\[formula\_l\]), respectively, and $\lambda \in [0,1]$ is the parameter which “turns on” anisotropy in momentum of the particle distribution. The functions $\phi$ and $\psi$ are assumed to fulfil the assumptions listed below. 