Existence of finite-order meromorphic solutions as a detector of integrability in difference equations

The existence of sufficiently many finite-order (in the sense of Nevanlinna) meromorphic solutions of a difference equation appears to be a good indicator of integrability. It is shown that, out of a large class of second-order difference equations, the only equation that can admit a sufficiently general finite-order meromorphic solution is the difference Painlevé II equation. The proof given relies on estimates obtained by arguments related to singularity confinement. The existence of meromorphic solutions of a general class of first-order difference equations is also proven by a simple method based on Banach’s fixed point theorem.