I tried:
NMinimize[Norm[((x^2-y^2 Cos[2/3 Sqrt[x^2-y^2]]) Cosh[y/3]) +
(y Sin[2/3 Sqrt[x^2-y^2]] Sinh[y/3])/Sqrt[x^2-y^2]-1], {x,y}]
and the result strongly hints that x==0 && y==0. Substituting those confirms this.
But you do have (x^2-y^2) and Sqrt[x^2-y^2] in denominators in your problem.