  We introduce the concept of a new kind of symmetric homeomorphisms on the
unit circle, which is derived from the generalization of symmetric
homeomorphisms on the real line. By the investigation of the barycentric
extension for this class of circle homeomorphisms and the biholomorphic
automorphisms induced by trivial Beltrami coefficients, we endow a complex
Banach manifold structure on the space of those generalized symmetric
homeomorphisms.
