One of the key ingredients of the recent classification of invariant

varieties for dynamical systems defined by coordinatwise actions of

polynomials, obtained by A. Medvedev and T. Scanlon, was a description of polynomial solutions of the functional equation A(X)=X(B). In our talk, using the theory of two-dimensional orbifolds, we present a description of solutions of this equation in arbitrary rational functions. As an

application, we prove an "effective'' version of the classical result of

Ritt about commuting rational functions.