  New types of irreducible second order Darboux transformations for the one
dimensional Schroedinger equation are described. The main feature of such
transformations is that the transformation functions have the eigenvalues
grater then the ground state energy of the initial (or reference) Hamiltonian.
When such a transformation is presented as a chain of two first order
transformations, an intermediate potential is singular and therefore
intermediate Hamiltonian can not be Hermitian while the final potential is
regular and the final Hamiltonian is Hermitian. Second derivative
supersymmetric quantum mechanical model based on a transformation of this kind
exhibits properties inherent to models with exact and broken supersymmetry at
once.
