  We study deformations of the quantum conformal mechanics of De
Alfaro-Fubini-Furlan with rational additional potential term generated by
applying the generalized Darboux-Crum-Krein-Adler transformations to the
quantum harmonic oscillator and by using the method of dual schemes and mirror
diagrams. In this way we obtain infinite families of isospectral and
non-isospectral deformations of the conformal mechanics model with special
values of the coupling constant $g=m(m+1)$, $m\in {\mathbb N}$, in the inverse
square potential term, and for each completely isospectral or gapped
deformation given by a mirror diagram, we identify the sets of the
spectrum-generating ladder operators which encode and coherently reflect its
fine spectral structure. Each pair of these operators generates a nonlinear
deformation of the conformal symmetry, and their complete sets pave the way for
investigation of the associated superconformal symmetry deformations.
