On elliptic Lax pairs and isomonodromic deformation systems for elliptic lattice equations

In a previous article [N. Delice, F.W. Nijhoff and S. Yoo-Kong, J. Phys. A: Math. Theor. 48(3) (2015), 035206] a novel class of elliptic Lax pairs for integrable lattice equations was introduced. The present article proposes a de-autonomisation of those Lax pairs leading to a class of elliptic discrete isomonodromic deformation problems. We analyse the systems of compatibility conditions using some (possibly novel) higher order elliptic identities.