Introduction

The problem of nuclear stability and of examining its various forms has become one of the hot issues in nuclear structure physics in view of the progress in construction of the large scale experimental facilities such as, e.g., GSI-Darmstadt (Germany), GANIL (France), or RIKEN (Japan), focusing on the properties of exotic nuclei. Some new avenues in this domain of research correspond to the exploitation of point-group symmetries - so far almost completely neglected in nuclear structure calculations, although extensively employed in other domains, notably in molecular physics. The arguments that deformed potentials may give rise to the Fermion-level degeneracies higher than 2 in realistic $N$-body systems like metallic clusters were formulated in Ref.[1]. The first results of realistic calculations applied to heavy nuclei of $A\sim230$ region, predicting the presence of the hypothetical tetrahedral minima together with exotic four-fold degeneracies, can be found in Ref.[2]. In the latter reference, the nuclear tetrahedral 'magic numbers' Z=56, 64, 70, and 90 and N=90, 112, and 136 were for the first time predicted. The tetrahedral symmetry instability has been studied in the light nuclei using self-consistent Hartree-Fock method.[3] A more complete survey of the tetrahedral magic numbers together with the group theoretical arguments about the research of the new forms of nuclear stability can be found in Ref.[4]. Some progress in relation to the description of the collective nuclear rotation using the methods adapted from those used in the molecular physics has been presented,[5] followed by a discussion of the problem of redundant variables in the nuclear quantum rotors.[6] A possibility of appearance of the pure octahedral symmetry nuclear configurations has also been discussed.[7] The possibility that some nuclei may be tetrahedral-symmetric in their ground-states[8] has been put forward in 2004. Very recently the arguments that the tetrahedral symmetry has already been seen in experiments were given,[9] while the first systematic Hartree-Fock-Bogolyubov calculations using different forces have been presented in Ref.[10]. Finally, the first investigation of the possible tetrahedral nuclear vibrations using the Generator Coordinate Method can be found in Ref.[11].