Hyperbolicity of covariant systems of first-order equations for vector and scalar fields

We consider a class of first-order systems of quasilinear partial differential equations $\dot{u}=L\text{'}[u,r]$, $\dot{r}=L{\text{'}}^{\text{'}}[u,r]$ that describe time evolution of the pair $⟨u,r⟩$ consisting of a vector field $u(x,t)$ and the set of scalar fields $r=⟨{\rho }^{\left(s\right)}(x,t);s=\mathrm{1,}\dots ,N⟩$, $x\in {ℝ}^3$. The class considered consists of systems that are invariant under time and space translations and covariant under space rotations. We describe the corresponding class of evolution generators, i.e., nonlinear first-order differential operators $L=⟨L\text{'}[\cdot ],L{\text{'}}^{\text{'}}[\cdot ]⟩$ acting in the functional space $C_{\mathrm{1,}}^{3+N}\left({ℝ}^3\right)$. Also, we find conditions under which a pair of operators $L$ generates a hyperbolic system.