Finite-order meromorphic solutions and the discrete Painleve equations

Let w(z) be an admissible finite-order meromorphic solution of the second-order difference equation w(z + 1) + w(z - 1) = R(z, w(z)) where R(z,w(z)) is rational in w(z) with coefficients that are meromorphic in z. Then either w(z) satisfies a difference linear or Riccati equation or else the above equation can be transformed to one of a list of canonical difference equations. This list consists of all known difference Painleve equations of the above form, together with their autonomous versions. This suggests that the existence of finite-order meromorphic solutions is a good detector of integrable difference equations.