New first integral for twisting type-N vacuum gravitational fields with two non-commuting Killing vectors

TL;DR

This paper introduces a new first integral for twisting type-N vacuum gravitational fields with two non-commuting Killing vectors, simplifying the problem to a complex second-order ODE and aiding in solving these equations.

Contribution

It presents a novel first integral and a reduction method for analyzing twisting type-N vacuum gravitational fields with non-commuting symmetries.

Findings

Derived a new first integral for the equations

Reduced the problem to a complex second-order ODE

Provided a method to solve the equations more efficiently

Abstract

A new first integral for the equations corresponding to twisting type-N vacuum gravitational fields with two non-commuting Killing vectors is introduced. A new reduction of the problem to a complex second-order ordinary differential equation is given. Alternatively, the mentioned first integral can be used in order to provide a first integral of the second-order complex equation introduced in a previous treatment of the problem.