# Geometry of B\"acklund Transformations I: Generality

## TL;DR

This paper uses Cartan's Method of Equivalence to analyze the generality and classification of rank-1 B"acklund transformations between hyperbolic Monge-Ampère systems, providing new examples and bounds.

## Contribution

It establishes an upper bound on the generality of generic rank-1 B"acklund transformations and classifies cases with symmetry groups of codimension 1, 2, or 3, introducing new auto-B"acklund examples.

## Key findings

- Upper bound for the generality of rank-1 B"acklund transformations
- Classification results for transformations with symmetry groups of codimension 1, 2, or 3
- New examples of auto-B"acklund transformations

## Abstract

Using Cartan's Method of Equivalence, we prove an upper bound for the generality of generic rank-1 B\"acklund transformations relating two hyperbolic Monge-Amp\`ere systems. In cases when the B\"acklund transformation admits a symmetry group whose orbits have codimension 1, 2, or 3, we obtain classification results and new examples of auto-B\"acklund transformations.