  We show that the potential in the radial equation in the model of coherent
quadrupole-octupole motion (CQOM) in nuclei generates a sequence of
superpotentials and subsequent series of effective potentials which satisfy the
shape-invariance condition and correspond to a SUSY-QM hierarchy of
Hamiltonians. On this basis we suggest that the original CQOM level scheme
possesses a generic supersymmetric structure of the spectrum inherent for the
coherent quadrupole-octupole mode. We outline the mechanism in which the real
quadrupole-octupole spectra in even-even and odd-even nuclei deviate from the
genuine symmetry. By using it we illustrate the possibilities to identify the
signs of supersymmetry in the alternating-parity spectra of even-even nuclei
and the quasi-parity-doublet levels of odd-mass nuclei described within the
CQOM model approach.
