Aequationes Math. 90 (2016), no. 5, 1025–1034*.* Also available on the arXiv. This is a joint work with John Osborne.

We study the class of functions meromorphic outside a countable closed set of essential singularities. We show that if a function in , with at least one essential singularity, permutes with a non-constant rational map , then is a Möbius map that is not conjugate to an irrational rotation. For a given function which is not a Möbius map, we show that the set of functions in that permute with is countably infinite. Finally, we show that there exist transcendental meromorphic functions such that, among functions meromorphic in the plane, permutes only with itself and with the identity map.