Killing Symmetries in $\mathcal{H}$-Spaces with $\Lambda$

TL;DR

This paper classifies all Killing symmetries in complex al-spaces with cosmological constant nd explicitly constructs metrics with null Killing vectors, reducing the problem to solving the Toda field equation.

Contribution

It provides a complete characterization of Killing symmetries in al-spaces with nd explicitly constructs metrics with null Killing vectors, linking the problem to the Toda equation.

Findings

All Killing symmetries in al-spaces with re identified.

Explicit metrics with null Killing vectors are derived.

The non-null Killing vector case reduces to solving the Toda field equation.

Abstract

All Killing symmetries in complex $\mathcal{H}$-spaces with $\Lambda$ in terms of the Pleba\'nski - Robinson - Finley coordinate system are found. All $\mathcal{H}$-metrics with $\Lambda$ admitting a null Killing vector are explicitly given. It is shown that the problem of non-null Killing vector reduces to looking for solution of the Boyer - Finley - Pleba\'nski (Toda field) equation