On the classification of multidimensionally consistent 3D maps

We classify multidimensionally consistent maps given by (formal or convergent) series of the following kind: $$T_k{x}_{ij}=x_{ij}+\sum _{m=2}^{\mathrm{\infty }}{A}_{ij;k}^{(m)}(x_{ij},x_{ik},x_{jk}),$$ where $A_{ij;, k}^{(m)}$arehomogeneouspolynomialsofdegree$m$ 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 $$T_k{x}_{ij}=\frac{x_{ij}+x_{ik}{x}_{jk}}{\sqrt{1-x_{ik}^2}\sqrt{1-x_{jk}^2}}\mathrm{.}$$