  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.
