Upon further inspection of the answers of my fellow colleagues I have realised I might have misunderstood what was meant by the toy model case. I thought this to be the three specific black bars added on the continuous plot but it makes more sense for the continuous plot to actually represent the classical case where every energy is attainable.
If then the question is “how to go from a finite amount of possible states to a continuum”, I would assume this could be done by filling up the space between the states of a single nucleus with those of others. This could be done by taking nuclei with different eta values and thus different deviations from the axial symmetry. In this manner we could attempt to “fill up” this gap between the states.