  Parameter-free theoretical predictions based on a dual shell mechanism within
the proxy-SU(3) symmetry of atomic nuclei, as well as covariant density
functional theory calculations using the DDME2 functional indicate that shape
coexistence (SC) based on the particle-hole excitation mechanism cannot occur
everywhere on the nuclear chart, but is restricted on islands lying within
regions of 7-8, 17-20, 34-40, 59-70, 96-112, 146-168 protons or neutrons.
Systematics of data for even-even nuclei possessing K=0 (beta) and K=2 (gamma)
bands support the existence of these islands, on which shape coexistence
appears whenever the K=0 bandhead 0_2^+ and the first excited state of the
ground state band 2_1^+ lie close in energy, with nuclei characterized by 0_2^+
lying below the 2_1^+ found in the center of these islands. In addition a
simple theoretical mechanism leading to multiple shape coexistence is briefly
discussed.
