Generalized factorization and isospectral potentials

It is shown that the method of using excited-state wave functions in constructing exactly solvable potentials as suggested by Berger and Ussembayev [Phys. Rev. A 82, 022121 (2010)] can be generalized further. Using this method, exactly solvable potentials related to the harmonic oscillator and some other potentials are obtained. The nonlinear symmetry algebras underlying the harmonic-oscillator models are also obtained explicitly.