This is the second paper on the path-integral approach of superintegrable systems on Darboux spaces, spaces of nonconstant curvature. We analyze in the spaces D_III and D_IV five and, respectively, four superintegrable potentials, which were first given by Kalnins et al. We are able to evaluate the path integral in most of the separating coordinate systems, leading to expressions for the Green functions, the discrete and continuous wave functions, and the discrete energy-spectra. In some cases, however, the discrete spectrum cannot be stated explicitly, because it is determined by a higher order polynomial equation. We also show that the free motion in Darboux space of type III can also contain bound states, provided the boundary conditions are appropriate. We can state the energy spectrum and the wave functions, respectively.

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