We consider a class of third-order evolution equations of the form \begin{ equation*} \left\{ \begin{ array}{ l} \displaystyle u_{ t}=F\left(x,t,u,u_x,u_{ xx},u_{ xxx},v,v_x,v_{ xx},v_{ xxx}\right), \displaystyle v_{ t}=G\left(x,t,u,u_x,u_{ xx},u_{ xxx},v,v_x,v_{ xx},v_{ xxx}\right), \end{ array} \right. \end{ equation*} describing pseudos-pherical (\textbf{ pss}) or spherical surfaces (\textbf{ 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 $\mathfrak{ g}=\mathfrak{ sl}(2,\R)$ or $\mathfrak{ g}=\mathfrak{ su}(2)$, when $K=-1$, $K=1$, respectively. We obtain characterization and also classification results. Applications of these results provide new examples and new families of such systems, which also contain systems of coupled KdV and mKdV-type equations and nonlinear Schr\"odinger equations. Additionally, this theory is applied to derive a B\"acklund transformation for the coupled KdV system.
Categories: math.DG math.AP