Integrability criteria for differential-difference systems: a comparison of singularity confinement and low-growth requirements

A new approach for the study of the integrability of differential-differenc e systems is introduced. Given a differential-difference system (of such a form that it can be iterated without necessitating the integration of a dif ferential equation at any step) a two-stage strategy is used. First, the si ngularity confinement necessary integrability criterion is used in order to limit the possible choices. Once the system is sufficiently reduced, the ( much stronger) requirement of nonexponential growth of the degree of the it erates of some initial condition is implemented. This method turns out to b e powerful and practical. The investigation of a given class of differentia l-difference systems has resulted in some well known integrable systems but also to two promising integrability candidates (one of which is reduced to a known integrable case).