Available experimental data indicates that the Kane models of the band structure of diamond-like semiconductors do not quite correctly describe anisotropy effects, and in particular, the nonparabolic character of the valence bands. In this paper, a uniform examination of the spectrum of all diamond-like semiconductors with a stype, lowest conduction band is presented in the framework of a generalization of both Kane four band models. A fourth-order dispersion equation (DE-4s) is obtained which permits one to state more precisely the limits of applicability of the Kane models for the realistic semiconductors under consideration. In the case of narrow gap semiconductors, the DE-4s transforms to a cubic DE possessing a number of advantages over the expression derived by the Kane theory. In this study it is shown that for a rather broad range of energies in the crystals examined the conduction and spinsplit hole bands are quite correctly described by Kane's theory (1957), that the heavy hole band is correctly described by Kane's equations (1956), but that for a description of the light hole bands it is necessary to employ the DE-4s equation.