  Supersymmetry transformations of first and second order are used to generate
Hamiltonians with known spectra departing from the harmonic oscillator with an
infinite potential barrier. It is studied also the way in which the
eigenfunctions of the initial Hamiltonian are transformed. The first and
certain second order supersymmetric partners of the initial Hamiltonian possess
third-order differential ladder operators. Since systems with this kind of
operators are linked with the Painlev\'e IV equation, several solutions of this
non-linear second-order differential equation will be simply found.
