Vector Hyperbolic Equations on the Sphere Possessing Integrable Third-Order Symmetries

The complete lists of vector hyperbolic equations on the sphere that have integrable third-order vector isotropic and anisotropic symmetries are presented. Several new integrable hyperbolic vector models are found. By their integrability, we mean the existence of vector Bäcklund transformations depending on a parameter. For all new equations, such transformations are constructed.