We give elementary proofs of two rigidity results. The first one asserts that the maximal entropy measure \mu_f of a rational map f is singular with respect to any given conformal measure excepted if f is a power, Tchebychev or Lattès map. This is a variation of a result of Zdunik. Our second result is an improvement of a theorem of Fisher and Urbański. It gives a sharp description of the exceptional functions that admit invariant line fields which are defined with respect to certain invariant measures