  We have recently proposed a strategy to produce, starting from a given
hamiltonian $h_1$ and a certain operator $x$ for which $[h_1,xx^\dagger]=0$ and
$x^\dagger x$ is invertible, a second hamiltonian $h_2$ with the same
eigenvalues as $h_1$ and whose eigenvectors are related to those of $h_1$ by
$x^\dagger$. Here we extend this procedure to build up a second hamiltonian,
whose eigenvalues are different from those of $h_1$, and whose eigenvectors are
still related as before. This new procedure is also extended to
crypto-hermitian hamiltonians.
