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A novel dual-shell mechanism for the phenomenon of shape coexistence innuclei within the Elliott SU(3) and the proxy-SU(3) symmetry is proposed forall mass regions. It is supposed, that shape coexistence is activated by largequadrupole-quadrupole interaction and involves the interchange among thespin-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 theharmonic 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 shapecoexistence may occur in certain islands on the nuclear map. The dual-shellmechanism 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 arepossible candidates for shape coexistence. In the light nuclei the nucleonsflip from the HO shell to the neighboring SO-like shell, which means, thatparticle excitations occur. For this mass region, the predicted islands ofshape coexistence, coincide with the islands of inversion. But in medium massand heavy nuclei, in which the nucleons inhabit the SO-like shells, shapecoexistence is accompanied by a merging of the SO-like shell with the open HOshell. 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 withthe SO-like shell.

The islands of shape coexistence within the Elliott and the proxy-SU(3) Models