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This is the second paper on the path integral approach of superintegrablesystems on Darboux spaces, spaces of non-constant curvature. We analyze in thespaces $\DIII$ and $\DIV$ five respectively four superintegrable potentials,which were first given by Kalnins et al. We are able to evaluate the pathintegral in most of the separating coordinate systems, leading to expressionsfor the Green functions, the discrete and continuous wave-functions, and thediscrete energy-spectra. In some cases, however, the discrete spectrum cannotbe stated explicitly, because it is determined by a higher order polynomialequation.  We show that also the free motion in Darboux space of type III can containbound states, provided the boundary conditions are appropriate. We state theenergy spectrum and the wave-functions, respectively

Path integral approach for superintegrable potentials on spaces of non-constant curvature: II. Darboux spaces D III and D IV