Two and three dimensional Hamiltonians with generalized and ordinary shape invariance
symmetry have been obtained by Fourier transforming over some coordinates
of the SU(3) Casimir operator defined on SUs3d/SUs2d symmetric space. It is
shown that the generalized shape invariance of the two dimensional Hamiltonian is
equivalent to SU(3) symmetry while in the three dimensional one, the ordinary
shape invariance is equivalent to contracted SU(3) and there is one to one correspondence
between the representations of the generalized shape invariance symmetry
of the two (three) dimensional Hamiltonian and SU(3) [contracted SU(3)]
Verma bases.