On generalized Dhombres equations with non-constant rational solutions in the complex plane

We characterize the generalized Dhombres functional equations f(zf(z))=φ(f(z)) which have a non-constant rational solution f0, holomorphic at 0, with f0(0)=w0, under the assumption that φ is a non-constant meromorphic function, holomorphic at w0. All rational solutions of such an equation which are holomorphic at 0 and all formal solutions are determined.