Null K\"ahler structures, Symmetries and Integrability

TL;DR

This paper reviews integrable systems from symmetry reductions of Plebanski's heavenly equations and shows how null Kähler-Einstein metrics with symmetry relate to a generalized dispersionless KP equation.

Contribution

It connects null Kähler-Einstein metrics with integrable systems via a generalized dispersionless KP equation, expanding understanding of symmetry reductions in four-dimensional geometries.

Findings

Null Kähler-Einstein metrics derived from generalized dispersionless KP equations.

Symmetry reductions of heavenly equations lead to integrable systems.

All four-dimensional null Kähler-Einstein metrics with symmetry can be obtained from these equations.

Abstract

We review the integrable systems which arise as symmetry reductions of Plebanski's heavenly equations, and their generalisations. We also show that all four-dimensional null Kahler-Einstein (or type N hyper-heavenly) metrics with symmetry can be found from solutions to a variable coefficient generalisation of the dispersionless Kadomtsev-Petviashvili equation.