# From the conformal self-duality equations to the Manakov-Santini system

## TL;DR

This paper shows how specific symmetry assumptions reduce four-dimensional anti-self-dual conformal equations to the three-dimensional Manakov-Santini system, linking complex geometric structures to integrable systems.

## Contribution

It explicitly derives reductions of anti-self-dual conformal structures to the Manakov-Santini system under two symmetry assumptions, expanding understanding of geometric-integrable system connections.

## Key findings

- Reduction under non-null translation yields the Manakov-Santini system.
- Reduction under homothety also leads to the Manakov-Santini system.
- Explores reductions related to null-Kähler conditions in both cases.

## Abstract

Under two separate symmetry assumptions, we demonstrate explicitly how the equations governing a general anti-self-dual conformal structure in four dimensions can be reduced to the Manakov-Santini system, which determines the three-dimensional Einstein-Weyl structure on the space of orbits of symmetry. The two symmetries investigated are a non-null translation and a homothety, which are previously known to reduce the second heavenly equation to the Laplace's equation and the hyper-CR system, respectively. Reductions on the anti-self-dual null-K\"ahler condition are also explored in both cases.