Exotic shape symmetries around the fourfold octupole magic number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>136</mml:mn></mml:mrow></mml:math> : Formulation of experimental identification criteria

We employ a realistic nuclear mean-field theory using the phenomenological, Woods-Saxon Hamiltonian with newly adjusted parameters containing no parametric correlations; originally present correlations are removed employing Monte Carlo approach. find very large neutron shell gaps at $N=136$ for all four octupole deformations ${\ensuremath{\alpha}}_{3\ensuremath{\mu}=0,1,2,3}$. These generate well-pronounced double potential-energy minima in standard multipole $({\ensuremath{\alpha}}_{20},{\ensuremath{\alpha}}_{22},{\ensuremath{\alpha}}_{3\ensuremath{\mu}},{\ensuremath{\alpha}}_{40})$ representation, often ${\ensuremath{\alpha}}_{20}=0$, which turn exotic symmetries ${C}_{2v}$, ${D}_{2d}$, ${T}_{d}$, and ${D}_{3h}$, discussed detail. The main goal of article is to formulate spectroscopic criteria experimental identification. Calculations macroscopic-microscopic method performed nuclei $Z\ensuremath{\ge}82$ $N\ensuremath{\ge}126$ multidimensional deformation spaces analyze expected shape instabilities mass table ``northeast'' doubly magic $^{208}\mathrm{Pb}$ nucleus. Whereas proton-unperturbed properties neutron-generated effects illustrated detail ${}_{Z=82}{\mathrm{Pb}}_{N&gt;126}$ nuclei, our discussion extended into even-even $Z&gt;82$ approaching less $Z/N$ ratios, encourage experiments could identify predicted symmetries. In addition tetrahedral point group symmetry, evidence has recently been published, we ${D}_{2d}$ symmetry resulting from superposition axially symmetric quadrupole two new symmetries, ${D}_{3h}$ associated ${\ensuremath{\alpha}}_{33}$ ${\ensuremath{\alpha}}_{31}$ energy minima, respectively. $n&gt;2$ treated as usual by projecting total potential energies onto $n=2$ subspace. Using representation groups quantum mechanical identification through analysis specific collective rotational bands generated band structures happen be markedly distinct structure ellipsoidal rotors; those various