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.Comment: 34 page
