Classification of Camassa-Holm-type differential systems describing pseudospherical or spherical surfaces

In this paper, we study nonlinear partial differential systems that describe surfaces of constant curvature. From the flatness condition of connection 1-forms, we present a classification of Camassa-Holm-type systems of the form

$$\{\begin{array}{rl}{u}_t-u_{xxt}& =F(x,t,u,u_x,\dots ,{\mathrm{\partial }}^{m}u/{\mathrm{\partial }}_x^m,v,v_x,\dots ,{\mathrm{\partial }}^{n}v/{\mathrm{\partial }}_x^n),\\ v_t-v_{xxt}& =G(x,t,u,u_x,\dots ,{\mathrm{\partial }}^{m}u/{\mathrm{\partial }}_x^m,v,v_x,\dots ,{\mathrm{\partial }}^{n}v/{\mathrm{\partial }}_x^n),\end{array}$$

with $m,n\ge 2$ and $F$, $G$ smooth functions, describing pseudospherical or spherical surfaces. We also establish classification results for a special type of third-order system. Applications of these results provide new examples of such systems, including the Song-Qu-Qiao system, the Xia-Qiao-Zhou system, and the two-component modified Camassa-Holm system. Furthermore, we construct the nonlocal symmetry for the Xia-Qiao-Zhou system from the gradients of the spectral parameter. By introducing an appropriate pseudo-potential, we prolong the nonlocal symmetry to an enlarged system and calculate the corresponding finite symmetry transformation. On this basis, we derive nontrivial solutions to the Xia-Qiao-Zhou system.