Bi-Hamiltonian structure of the oriented associativity equation

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.