Hyperbolic first-order covariant evolution equations for vector fields in $\mathbb{R}^3$

The class $\mathfrak{K}_1(\mathbb{R}^3)$ of systems of first-order quasilinear partial differential equations is considered. Such systems $\dot{\boldsymbol{u}}=\mathsf{L}[\boldsymbol{u}]$ describe the evolution of vector fields $\boldsymbol{u}(\boldsymbol{x},t)$, $\boldsymbol{x}\in\mathbb{R}^3$ in time $t\in\mathbb{R}$. The class $\mathfrak{K}_1(\mathbb{R}^3)$ consists of all systems that are invariant under translations in time $t\in\mathbb{R}$ and space $\mathbb{R}^3$ and are covariant under rotations of $\mathbb{R}^3$. We describe the class of first-order nonlinear differential operators $\mathsf{L}$ acting in the functional space $C_{1,\operatorname{loc}}(\mathbb{R}^3)$ that are evolution generators of such systems. We obtain a necessary and sufficient condition for the operator $\mathsf{L}\in\mathfrak{K}_1(\mathbb{R}^3)$ to generate a hyperbolic system.