# Bi-Hamiltonian structure of the Oriented Associativity Equation

## TL;DR

This paper demonstrates that the Oriented Associativity equation, important in Integrable Systems, possesses a bi-Hamiltonian structure with both local and non-local Hamiltonian operators, revealing deep algebraic geometric connections.

## Contribution

It proves the existence of a third-order non-local Hamiltonian operator for the equation, enriching the understanding of its integrable structure.

## Key findings

- The equation is Hamiltonian with a first-order operator.
- It admits a third-order non-local homogeneous Hamiltonian operator.
- This provides a new example linking integrable systems and algebraic geometry.

## Abstract

The Oriented Associativity equation plays a fundamental role in the theory of Integrable Systems. In this paper we prove that the equation, besides being Hamiltonian with respect to a first-order Hamiltonian operator, has a third-order non-local homogeneous Hamiltonian operator belonging to a class which has been recently studied, thus providing a highly non-trivial example in that class and showing intriguing connections with algebraic geometry.