Dunajski-Tod equation and reductions of the generalized dispersionless 2DTL hierarchy

TL;DR

This paper explores reductions of the generalized dispersionless 2DTL hierarchy, showing that the simplest reduction leads to the Dunajski-Tod equation, which describes ASD vacuum metrics with conformal symmetry.

Contribution

It extends the reduction scheme to the two-component dispersionless 2DTL hierarchy and links the simplest reduction to the Dunajski-Tod equation.

Findings

Simplest reduction yields the Dunajski-Tod equation

Higher reductions lead to new hierarchies

Connection to ASD vacuum metrics with conformal symmetry

Abstract

We transfer the scheme for constructing differential reductions recently developed for the Manakov-Santini hierarchy to the case of the two-component generalization of dispersionless 2DTL hierarchy. We demonstrate that the equation arising as a result of the simplest reduction is equivalent (up to a Legendre type transformation) to the Dunajski-Tod equation, locally describing general ASD vacuum metric with conformal symmetry. We consider higher reductions and corresponding reduced hierarchies also.