Department of Mathematics,

Brock University,

St Catharines,

Ontario L2S 3A1,

CANADA

E-mail: sanco@brocku.ca

**Symmetry-integrability classifications of hyperbolic
vector evolution equations**

**Abstract:**

Motivated by recent work on integrable flows of curves and 1+1 dimensional
sigma models, some symmetry-integrability classifications of $O(N)$-invariant

hyperbolic equations $U_{tx} =f(U,U_t,U_x)$ for an $N$-component vector
$U(t,x)$ are presented. In each class all scaling-homogeneous equations
admitting a higher-symmetry of least possible scaling weight are obtained
by computer-algebra computations. Sigma model interpretations of these
equations are used to derive

bi-Hamiltonian formulations giving a proof of integrability.