equation (\[eq\_vlasov\]) can be dealt with by the method of characteristics which is now described.

\[lem\_conserved\_quan\] The quantities $E$ and $L$, defined on the tangent bundle $T \mathscr M$, by $$\begin{aligned}
L &:= \varrho H e^{ -\gamma\nu} v^3 - q A_\varphi, \label{formula_l} \\
E &:= \frac{e^{\gamma\nu} v^0 - 1}{\gamma} + \omega \varrho H e^{ -\gamma\nu} v^3 + q A_t, \label{formula_e}\end{aligned}$$ are conserved along the characteristic curves of the Vlasov equation, i.e. $$\mathfrak TE = 0, \quad \mathfrak TL = 0.$$

The assertion of this lemma can be shown via a direct calculation and it is moved to the appendix.

Unlike the uncharged case, in the charged case the characteristic curves of the Vlasov equations are not the lifts of the geodesics to $T\mathscr M$. Consequently the conserved quantities cannot be obtained by $g(X, p)$, where $X$ is a Killing vector field and $p$ is the canonical momentum. However, this structure can still be recognised in the present case. If we define $$\begin{aligned}
\tilde E &:= -g(\partial_t, p), \label{tilde_e} \\
\tilde L &:= g(\partial_\varphi, p), \label{tilde_l}\end{aligned}$$ it turns out that the quantities $E$ and $L$ can be obtained from $\tilde E$ and $\tilde L$ by taking into account a suitable correction 