On the classification of multidimensionally consistent 3D maps

TL;DR

This paper classifies a specific class of multidimensionally consistent 3D maps and finds that the well-known symmetric discrete Darboux system is the only non-trivial example within this class.

Contribution

It provides a complete classification of a class of 3D maps, showing the symmetric discrete Darboux system as the unique non-trivial solution.

Findings

The symmetric discrete Darboux system is uniquely characterized as the only non-trivial map in the class.

The classification applies to maps expressed as formal or convergent series with polynomial coefficients.

The result deepens understanding of integrable systems and their discrete geometric structures.

Abstract

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