# Geometry of B\"acklund Transformations II: Monge-Amp\`ere Invariants

## TL;DR

This paper investigates the existence of rank-1 Bäcklund transformations between hyperbolic Euler-Lagrange systems in the plane, identifying obstructions via local invariants and discovering new transformations relating systems of different types.

## Contribution

It introduces a framework to determine when Bäcklund transformations exist between hyperbolic Euler-Lagrange systems and uncovers a novel class relating systems of distinct types.

## Key findings

- Identified obstructions to Bäcklund transformations using local invariants.
- Established conditions for the existence of rank-1 Bäcklund transformations.
- Discovered a new class of transformations connecting systems of different types.

## Abstract

This article is concerned with the question: For which pairs of hyperbolic Euler-Lagrange systems in the plane does there exist a rank-$1$ B\"acklund transformation relating them? We express some obstructions to such existence in terms of the local invariants of the Euler-Lagrange systems. In addition, we discover a class of B\"acklund transformations relating two hyperbolic Euler-Lagrange systems of distinct types.