The elliptic lattice KdV system revisited

The elliptic lattice KdV system, discovered in 2003, is an extension of the lattice potential KdV equation associated with an elliptic curve. This is a rather complicated three-component system on the quad lattice, which contains the moduli of the elliptic curve as parameters. In this paper, we investigate this system further and, among other results, derive a two-component multiquartic form of the system on the quad lattice. Furthermore, we construct an elliptic Yang–Baxter map and study the associated continuous and semidiscrete systems. In particular, we derive the so-called “generating PDE” for this system, comprising a six-component system of second-order PDEs, which can be considered to constitute an elliptic extension of the Ernst equations of General Relativity.