Volume preserving multidimensional integrable systems and Nambu-Poisson Geometry

TL;DR

This paper explores a class of volume-preserving multidimensional integrable systems using Nambu-Poisson mechanics, extending twistor constructions and relating them to dispersionless KP equations and self-dual Einstein equations.

Contribution

It introduces a generalized twistor construction for volume-preserving integrable systems within Nambu-Poisson geometry, linking them to dispersionless KP and self-dual Einstein equations.

Findings

Established a twistor construction for the new integrable systems.

Connected these systems to dispersionless KP and self-dual Einstein equations.

Extended the Gindikin's pencil of two forms to Nambu-Poisson systems.

Abstract

In this paper we study generalized classes of volume preserving multidimensional integrable systems via Nambu-Poisson mechanics. These integrable systems belong to the same class of dispersionless KP type equation. Hence they bear a close resemblance to the self dual Einstein equation. Recently Takasaki-Takebe provided the twistor construction of dispersionless KP and dToda type equations by using the Gindikin's pencil of two forms. In this paper we generalize this twistor construction to our systems.