Mutations of potentials

This is a joint work with A.Usnich. What are nontrivial collections of regular functions (on two-dimensional torus) such that any two of them are birational but not biregular conjugate? Our lemma builds infinite collections of this kind from initial function satisfying some conditions. There are 10 examples related to del Pezzo surfaces. In the case of the plane our functions are associated with solutions of Markov's equation.