Poincare group of symmetry leads to Klein-Gordon equation

Run NTMF064.Package.m first!

Clear["Global`*"]

Is the Klein-Gordon equation the only linear scalar equation of the second order which is invariant under the Poincare group of transformations?

(* Independent variables *)
IndepVar={x[0],x[1],x[2],x[3]};
ivar=Delete[IndepVar,0];
(* Dependent variables *)
DepVar={ψ};
(* PDE *)
PDEs = {w ψ[ivar]+Sum[a[k] D[ψ[ivar],x[k]],{k,0,3}]+Sum[Sum[If[k>l,0,b[k,l] D[ψ[ivar],x[k],x[l]]],{l,0,3}],{k,0,3}]}

Expression to substitute for in the infinitesimal criterion of invariance

subs = {ψ[ivar]};
sol=Solve[PDEs==0, subs]

Infinitesimals of the Poincare group and infinitesimal criterion of invariance

nc=10;
ξ[x[0]] = c[7]+c[4]x[1]+c[5]x[2]+c[6]x[3];
ξ[x[1]] = c[8]+c[2]x[3]-c[3]x[2]+c[4]x[0];
ξ[x[2]] = c[9]-c[1]x[3]+c[3]x[1]+c[5]x[0];
ξ[x[3]] = c[10]+c[1]x[2]-c[2]x[1]+c[6]x[0];η[ψ] =0;
zero=CheckPointSymmetryOfDE[PDEs,subs, IndepVar, DepVar,ξ,η]

If it is not zero then find the equations for coefficients

variables=Flatten[Table[ψ[j],{j,0,14}]] ;
variables=Flatten[Union[variables,Table[c[j],{j,1,nc}]]]
Column[GetConditionsForPointSymmetries[zero,variables]]

Infinitesimal generators, point transformations and commutator table from the last ansatz

ShowPointSymmetriesAndCommutationRelations[X,f,ϵ,IndepVar,DepVar,ξ,η,c,10,{}]