# Integrability properties of symmetric 4+4-dimensional heavenly type equation

## TL;DR

This paper extends the dispersionless $ar ext{ extbackslash d}$-dressing method to symmetric heavenly equations in 4+4 and higher dimensions, deriving solutions and reduction techniques for these integrable systems.

## Contribution

It introduces a dressing scheme for symmetric heavenly equations in higher dimensions and demonstrates reduction to four-dimensional equations at the dressing data level.

## Key findings

- Developed a dressing scheme for 4+4-dimensional heavenly equations.
- Derived a class of special solutions for these equations.
- Showed effective reduction from higher to lower dimensions.

## Abstract

We demonstrate that the dispersionless $\bar\partial$-dressing method developed before for general heavenly equation is applicable to the $4+4$ and $2N+2N$ - dimensional symmetric heavenly type equations. We introduce generating relation and derive the two-form defining the potential and equation for it. We develop the dressing scheme, calculate a class of special solutions and demonstrate that reduction from $4+4$-dimensional equation to four-dimensional general heavenly equation can be effectively performed on the level of the dressing data. We consider also the extension of the proposed scheme to $2N+2N$-dimensional case.