  A novel dual-shell mechanism for the phenomenon of shape coexistence in
nuclei within the Elliott SU(3) and the proxy-SU(3) symmetry is proposed for
all mass regions. It is supposed, that shape coexistence is activated by large
quadrupole-quadrupole interaction and involves the interchange among the
spin-orbit (SO) like shells within nucleon numbers 6-14, 14-28, 28-50, 50-82,
82-126, 126-184, which are being described by the proxy-SU(3) symmetry, and the
harmonic oscillator (HO) shells within nucleon numbers 2-8, 8-20, 20-40, 40-70,
70-112, 112-168 of the Elliott SU(3) symmetry. The outcome is, that shape
coexistence may occur in certain islands on the nuclear map. The dual-shell
mechanism predicts without any free parameters, that nuclei with proton number
(Z) or neutron number (N) between 7-8, 17-20, 34-40, 59-70, 96-112, 146-168 are
possible candidates for shape coexistence. In the light nuclei the nucleons
flip from the HO shell to the neighboring SO-like shell, which means, that
particle excitations occur. For this mass region, the predicted islands of
shape coexistence, coincide with the islands of inversion. But in medium mass
and heavy nuclei, in which the nucleons inhabit the SO-like shells, shape
coexistence is accompanied by a merging of the SO-like shell with the open HO
shell. The shell merging can be accomplished by the outer product of the SU(3)
irreps of the two shells and represents the unification of the HO shell with
the SO-like shell.
