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