Continuous symmetric reductions of the Adler–Bobenko–Suris equations

Continuously symmetric solutions of the Adler–Bobenko–Suris class of discrete integrable equations are presented. Initially defined by their invariance under the action of both of the extended three-point generalized symmetries admitted by the corresponding equations, these solutions are shown to be determined by an integrable system of partial differential equations. The connection of this system to the Nijhoff–Hone–Joshi 'generating partial differential equations' is established and an auto-Bäcklund transformation and a Lax pair for it are constructed. Applied to the H1 and Q1?=0 members of the Adler–Bobenko–Suris family, the method of continuously symmetric reductions yields explicit solutions determined by the Painlevé trancendents.