# Dispersionless Multi-Dimensional Integrable Systems and Related Conformal Structure Generating Equations of Mathematical Physics

## TL;DR

This paper investigates multi-dimensional dispersionless integrable systems linked to conformal structures in mathematical physics, using Lie-algebraic methods and analyzing various heavenly and Einstein-Weyl equations to establish their integrability.

## Contribution

It introduces a modified Lie-algebraic approach for proving integrability of conformal structure equations and explores their multi-dimensional generalizations and superconformal analogs.

## Key findings

- Proved complete integrability of several conformal structure equations.
- Analyzed multi-dimensional generalizations of heavenly and Einstein-Weyl equations.
- Constructed superconformal analogs of Whitham heavenly equations.

## Abstract

Using diffeomorphism group vector fields on $\mathbb{C}$-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of mathematical physics. An interesting modification of the devised Lie-algebraic approach subject to spatial-dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied in proving complete integrability of some conformal structure generating equations. As examples, we analyze the Einstein-Weyl metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations and the inverse first Shabat reduction heavenly equation. We also analyze the modified Pleba\'nski heavenly equations, the Husain heavenly equation and the general Monge equation along with their multi-dimensional generalizations. In addition, we construct superconformal analogs of the Whitham heavenly equation.