We study the supersymmetric partners of the harmonic oscillator with an
infinite potential barrier at the origin and obtain the conditions under which
it is possible to add levels to the energy spectrum of these systems. It is
found that instead of the usual rule for non-singular potentials, where the
order of the transformation corresponds to the maximum number of levels which
can be added, now it is the integer part of half the order of the
transformation which gives the maximum number of levels to be created.Comment: This article is a natural continuation of arXiv:1603.05173 and
arXiv:1310.562
