On the classification of multidimensionally consistent 3D maps

Abstract

We classify multidimensionally consistent maps given by (formal or convergent) series of the following kind:

$$\begin{aligned} T_k x_{ij}=x_{ij} + \sum _{m=2}^\infty A_{ij ; \, k}^{(m)}(x_{ij},x_{ik},x_{jk}), \end{aligned}$$

where $A_{ij;\, k}^{(m)}$ are homogeneous polynomials of degree m of their respective arguments. The result of our classification is that the only non-trivial multidimensionally consistent map in this class is given by the well-known symmetric discrete Darboux system

$$\begin{aligned} T_k x_{ij}=\frac{x_{ij}+x_{ik}x_{jk}}{\sqrt{1-x_{ik}^2}\sqrt{1-x_{jk}^2}}. \end{aligned}$$