we write in a flux-conservative form as

$$\begin{aligned}
\begin{split}
\partial_t (\sqrt{\gamma} B^i)+\partial_k(-\beta^k \sqrt{\gamma} B^i
+ \alpha \epsilon^{ikj} \sqrt{\gamma} E_j) =
  -\sqrt{\gamma} B^k
(\partial_k \beta^i) - \alpha \sqrt{\gamma} \gamma^{ij} \partial_j
\phi \,, 
\end{split}
\label{magnetic_evol}
\\
\begin{split}
\partial_t (\sqrt{\gamma} E^i) + \partial_k(-\beta^k \sqrt{\gamma} E^i
- \alpha \epsilon^{ikj} \sqrt{\gamma} B_j) = 
  -\sqrt{\gamma} E^k
(\partial_k \beta^i) - \alpha \sqrt{\gamma} \gamma^{ij} \partial_j \psi - 
\alpha \sqrt{\gamma} J^i\,,
\end{split}
\label{electric_evol}
\\
\begin{split}
\partial_t \phi + \partial_k (-\beta^k \phi + \alpha B^k) =
  -\phi
(\partial_k \beta^k) + B^k (\partial_k \alpha) - \frac{\alpha}{2}
(\gamma^{lm} \partial_k \gamma_{lm}) B^k - \alpha \kappa \phi\,,
\end{split}
\label{psib_evol}
\\
\begin{split}
 \partial_t \psi + \partial_k (-\beta^k \psi + \alpha E^k) =
   -\psi
 (\partial_k \beta^k) + E^k (\partial_k \alpha) - \frac{\alpha}{2}
 (\gamma^{lm} \partial_k \gamma_{lm}) E^k  + \alpha q - \alpha \kappa
 \psi\,,
\end{split}
\label{eq:psie_evol}
\\
\begin{split} 
\partial_t (\sqrt{\gamma} q) + \partial_k [\sqrt{\gamma} (-\beta^k q
 + \alpha J^k)] = 0\,,
\end{split}
\label{charge_evol}
\\
\begin{split} \partial_t (\sqrt{\gamma} D) + \partial_k [\sqrt{\gamma} (-\beta^k D
 + \alpha v^k D)] = 0\,,
\end{split}
\label{rest_mass_evol}
\\
\begin{split} \partial_t (\sqrt{\gamma} \tau) + \partial_k \{\sqrt{\gamma}
 [-\beta^k \tau + \alpha ( S^k - v^k D)]\} =
   \sqrt{\gamma} (\alpha
 S^{lm} K_{lm} - S^k \partial_k \alpha)\,,
\end{split}
\label{energy_density_evol}
\\
\begin{split}
\partial_t (\sqrt{\gamma} S_i) + \partial_k [\sqrt{\gamma}
  