Islands of oblate hyperdeformed and superdeformed superheavy nuclei with D3h point group symmetry in competition with normal-deformed D3h states: "Archipelago" of D3h-symmetry islands

In two recent articles we have formulated nuclear mean-field theory predictions of existence of a new form of magic numbers, referred to as fourfold magic numbers. These predictions stipulate the presence of strong shell closures at the neutron numbers N = 136 (actinide region) and N = 196 (superheavy region) simultaneously at nonvanishing all four octupole deformations alpha 3 mu=0,1,2,3 not equal 0. In contrast to the traditional notion of magic numbers, the new notion refers to simultaneous nonspherical configurations (alpha 3 mu not equal 0, alpha 2 mu = 0). In this article we study the nuclear equilibrium deformations with alpha 33 not equal 0 combined with nonvanishing quadrupole deformation, alpha 20 not equal 0. One easily shows that such geometrical shapes have a threefold symmetry axis and are invariant under the symmetry operations of the D3h point group. We employ a realistic phenomenological mean-field approach with the so-called universal deformed Woods-Saxon potential and its recently optimized parametrization based on actualized experimental data with the help of the inverse problem theory methods. The presence of parametric correlations among 4 of 12 parameters in total was detected and removed employing Monte Carlo approach leading to stabilization of the modeling predictions. Our calculations predict the presence of three nonoverlapping groups of nuclei with D3h symmetry, referred to as islands on the nuclear (Z, N) plane (mass table). These islands lie in the rectangle 110 << Z << 138 and 166 << N << 206. The "repetitive" structures with the D3h symmetry minima are grouped in three zones of oblate quadrupole deformation, approximately, at alpha 20 e [-0.10, -0.20] (oblate normal deformed), around alpha 20 similar to -0.5 (oblate superdeformed) and alpha 20 similar to -0.85 (oblate hyperdeformed). Importantly, the energies of those latter exotic deformation minima are predicted to be very close to the ground-state energies. We illustrate, compare, and discuss the evolution of the underlying shell structures. Nuclear surfaces parametrized as usual with the help of real deformation parameters, {alpha lambda mu = alpha*lambda mu], are invariant under Oxz-plane reflection, the symmetry also referred to as y simplex (Sy). For the shapes with odd-multipolarity (lambda -> lambda odd = 3, 5, 7, ...) it follows that E(-alpha lambda odd,mu) = E(+alpha lambda odd,mu). It turns out that the predicted equilibrium deformations generate symmetric double (or "twin") minima separated by potential barriers, whose heights vary with the nucleon numbers, possibly inducing the presence of parity-doublets in the spectra. To facilitate possible experimental identification of such structures, we examine the appearance of such doublets solving the collective Schrodinger equation. Implied suggestions are illustrated and discussed.