An infinite hierarchy of symmetries associated with hyperbolic surfaces

A subclass of hyperbolic surfaces defined by the requirement that the negative Gaussian curvature be a function of one asymptotic coordinate only is considered. Thus, these Bianchi surfaces generalize pseudo-spherical surfaces governed by the classical sine-Gordon equation. It is shown that the modified Korteweg-de Vries (MKDV) hierarchy associated with the latter equation may be used to construct an infinite hierarchy of coordinate-dependent symmetries of the corresponding Bianchi system. Accordingly, an extension of the well-known MKDV recursion operator is presented.