Darboux related quantum integrable systems on a constant curvature surface

We consider integrable deformations of the Laplace–Beltrami operator on a constant curvature surface, obtained through the action of first-order Darboux transformations. Darboux transformations are related to the symmetries of the underlying geometric space and lead to separable potentials which are related to the KdV equation. Eigenfunctions of the corresponding operators are related to highest weight representations of the symmetry algebra of the underlying space.