We examine the properties of the wave functions which are related to the fine structure of the Landau levels of crystalline electrons. Within a recently proposed formulation we are able to calculate explicit wave functions for a special periodic potential and a certain rotional field. The wave functions turn out to be highly singular in terms of the degeneracy parameter ky of the Landau free-electron functions. We show how the singularities can be removed to obtain physically reasonable results. In addition, we discuss transformation properties of the wave functions with respect to rotations of the reference frame connected to the free-electron Landau functions. Also we come to an agreement with Brown and Meisel on the degeneracy of the diamagnetic fine-structure levels.