# Group invariant transformations for the Klein-Gordon equation in three dimensional flat spaces

## TL;DR

This paper classifies all potential functions for the Klein-Gordon equation in three-dimensional flat spaces that admit Lie and Noether symmetries, linking symmetries to the conformal algebra of the geometry and deriving invariant solutions.

## Contribution

It provides a complete symmetry classification of the Klein-Gordon equation in symmetric spacetimes, identifying potentials with specific symmetry properties and solutions.

## Key findings

- Classified all potentials $V(t,x,y)$ admitting Lie and Noether symmetries.
- Derived invariant solutions for certain symmetric potentials.
- Linked symmetries of the equation to the conformal algebra of the underlying geometry.

## Abstract

We perform the complete symmetry classification of the Klein-Gordon equation in maximal symmetric spacetimes. The central idea is to find all possible potential functions $V(t,x,y)$ that admit Lie and Noether symmetries. This is done by using the relation between the symmetry vectors of the differential equations and the elements of the conformal algebra of the underlying geometry. For some of the potentials, we use the admitted Lie algebras to determine corresponding invariant solutions to the Klein-Gordon equation. An integral part of this analysis is the problem of the classification of Lie and Noether point symmetries of the wave equation.