On a class of systems of hyperbolic equations describing pseudo-spherical or spherical surfaces

We consider systems of partial differential equations of the form \begin{equation}\nonumber \left\{ \begin{array}{l} u_{xt}=F\left(u,u_x,v,v_x\right),\\ v_{xt}=G\left(u,u_x,v,v_x\right), \end{array} \right. \end{equation} describing pseudospherical (pss) or spherical surfaces (ss), meaning that, their generic solutions $u(x,t)\, v(x,t)$ provide metrics, with coordinates $(x,t)$, on open subsets of the plane, with constant curvature $K=-1$ or $K=1$. These systems can be described as the integrability conditions of $\mathfrak{g}$-valued linear problems, with $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{R})$ or $\mathfrak{g}=\mathfrak{su}(2)$, when $K=-1$, $K=1$, respectively. We obtain characterization and also classification results. Applications of the theory provide new examples and new families of systems of differential equations, which contain generalizations of a Pohlmeyer-Lund-Regge type system and of the Konno-Oono coupled dispersionless system.