  In this article we will apply the first- and second-order supersymmetric
quantum mechanics to obtain new exactly-solvable real potentials departing from
the inverted oscillator potential. This system has some special properties; in
particular, only very specific second-order transformations produce
non-singular real potentials. It will be shown that these transformations turn
out to be the so-called complex ones. Moreover, we will study the factorization
method applied to the inverted oscillator and the algebraic structure of the
new Hamiltonians.
