Local Yang–Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation

We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov equation, and their matrix Lax representations defined by local Yang--Baxter equation. Sergeev [S.M. 1998 Lett. Math. Phys. 45, 113--119] presented classification results on three-dimensional maps obtained from equation for a certain class of matrix-functions in situation when possesses unique solution determines map. In this paper, using correspondences arising some simple $2\times 2$ matrix-functions, we show that there (non-unique) define do not belong list; paves way new, wider maps. present invariants derived prove Liouville integrability them. Furthermore, approach solving obtain several new birational including with arbitrary groups, $9$-dimensional map associated Darboux transformation derivative nonlinear Schr\"odinger (NLS) generalisation $3$-dimensional Hirota