For pt.I see ibid., vol.5, p.783 (1989). The authors study possible transformations producing isospectral Hamiltonians under one united framework-the application of the commutation formula through an inverse scattering formulation. In addition to the four cases of such transformations, suitable compositions of which make up all isospectral transformations known under the title of supersymmetric quantum mechanics, they introduce two new one-parameter sets of such transformations. These two new 'isospectral' transformations generate Hamiltonians with singular-soliton potentials. Three types of eigenstates can be defined for such a class of Hamiltonians, the 'normal' bound state, the singular bound state and the null bound state, characterised by positive, negative and zero norm respectively. Various compositions of the transformations are also analysed.