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 $\dot{\boldsymbol{u}}=\mathsf{L}'[\boldsymbol{u},\boldsymbol{\rho}]$, $\dot{\boldsymbol{\rho}}=\mathsf {L}''[\boldsymbol{u},\boldsymbol{\rho}]$ that describe time evolution of the pair $\langle\boldsymbol{u},\boldsymbol{\rho}\rangle$ consisting of a vector field $\boldsymbol{u}(\boldsymbol{x},t)$ and the set of scalar fields $\boldsymbol{\rho}=\langle\rho^{(s)}(\boldsymbol{x},t);\ s=1,\dots,N\rangle$, $\boldsymbol{x}\in\mathbb{R}^3$. 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 $\mathsf{L}=\langle\mathsf{L}'[\cdot],\mathsf{L}''[\cdot]\rangle$ acting in the functional space $C_{1,\mathrm{loc}}^{3+N}(\mathbb{R}^3)$. Also, we find conditions under which a pair of operators $\mathsf{L}$ generates a hyperbolic system.