Some symmetry classifications of hyperbolic vector evolution equations

Motivated by recent work on integrable flows of curves and 1+1 dimensional sigma models, several O(N)-invariant classes of hyperbolic equations $u_{tx} =f(u,u_t,u_x)$ for an $N$-component vector $u(t,x)$ are considered. In each class we find all scaling-homogeneous equations admitting a higher symmetry of least possible scaling weight. Sigma model interpretations of these equations are presented.