Darboux transformations with tetrahedral reduction group and related integrable systems

In this paper we derive new two-component integrable differential difference and partial difference systems by applying a Lax-Darboux scheme to an operator formed from an ${\mathfrak{s}\mathfrak{l}}_3(\mathbb{C})$-based automorphic Lie algebra. The integrability of the found systems is demonstrated via Lax pairs and generalised symmetries.