It is more than ten years by now that the study of superdeformed (SD) shapes in nuclei constitute one of the main venues in nuclear spectroscopy. Today it is well understood that an increased stability of strongly elongated nuclei results from the quantum (shell) effects that manifest themselves, among others, through a lowering of the nucleonic level densities at certain nucleon numbers. Within the anisotropic harmonic oscillator (HO) model, such shell effects arise when the ratios among the three principal frequencies are equal to the ratios of small integer numbers. The strongest shell closures correspond to axially deformed nuclei with the semi-axis ratios 2:1:1 and 3:2:2, the axis ratios being simply related to the oscillator frequencies .
The HO model is of course only poor an approximation for the majority of nuclei, for which the spin-orbit interactions play a determining role. Yet as it happens, the nuclear mean field obeys approximately a specific SU(2) symmetry, usually referred to as a pseudo-spin. (For an early formulation of the pseudo-spin symmetry see Refs. [2,3]; the contemporary formulation of the problem is based on the Dirac formalism according to the scheme proposed in Ref.  and further developed recently in [5,6].) Taking into account this symmetry allows at the same time to take care of the strong spin-orbit coupling, and profit from the simplicity of the HO model. Indeed, basing on the pseudo-spin symmetry, and employing a realistic deformed mean field Hamiltonian, it was possible to predict  (after the initial discovery of the SD band in 152Dy  but several years before the experiments in other regions have been done), the existence of the whole groups of SD nuclei. Moreover, the predictions gave also the fact that the deformations of strongly elongated shapes may considerably deviate from the 2:1:1 HO rule; these deviations are now confirmed through numerous experiments. The abundance scheme for the nuclear SD states at high angular momenta is well established today in the so-called A190, A150, A130, and A80 regions, see Refs. [9,10,11,12] for reviews; it includes also the recently discovered SD states in the A60 region [13,14,15,16,17,18,19,20], as well as a region of fission isomers in A240 nuclei, known already for a long time but at relatively low angular momenta.
Numerous cluster structures in light nuclei can also be interpreted as SD states, see Refs. [21,22] for more details and a more exhaustive reference list. In particular, the 2:1:1 deformed HO model predicts the SD shell-closures  at particle numbers 2, 4, 10, 16 and 28, a sequence characterized by an increased stability at large deformations, and also compatible with the -cluster structures. This gives, for example, the - cluster ground state of 8Be, or the 16O- cluster state in 20Ne. Prolonging the same sequence, one may expect stable SD structures in 26Ne and 32S. Next, doubly-magic SD states should appear at N=Z=28 (not to be confused with the spherical shell closures at the same nucleon numbers). However, because of the increasing role of the spin-orbit interaction when the nucleonic numbers increase, these values are slightly modified. This gives the doubly-magic SD nucleus 60Zn at the center of the experimentally known SD A60 region.
One can see that the SD states in 32S (although experimentally not discovered to date) constitute a missing link between the known cluster SD states in very light nuclei, and the known SD states in the A60 region. On the one hand, the first indications that the cluster SD states in 32S may exist are provided by the measurements in the 16O-16O breakup channel , and by the 16O-16O molecular resonances, as quoted in Ref. . On the other hand, several mean-field calculations, both non-self-consistent  and self-consistent [26,27,28,29], as well as the -cluster calculations [30,24], predict in 32S an existence of the 2:1:1 deformed structures. It is not clear at the moment, what exactly is the relationship between the molecular states (a pair of touching 16O nuclei), and the SD states (a compact matter distribution), although both classes may correspond to the same axis ratios and deformations. Such strongly deformed states should coexist with numerous low-deformation states already known in this nucleus . In fact, the latter ones are very well described by the sd-shell model .
One may expect a number of interesting physical phenomena that can be studied in the hypothetical SD configurations of 32S, such as the shape coexistence, competition between various decay channels, proton neutron pairing and its deformation dependence, effects related to the time-odd components of nuclear mean fields, as well as nuclear-molecular and nuclear-cluster structures. Detailed properties may be significantly influenced by the presence of intruder states originating from the N0=3, and even N0=4 HO shells. With a total number of nucleons strongly restrained (only 16 per one kind of particles) one should expect a pronounced variation of shapes from one single-particle (particle-hole) configuration to another.
In the present paper we aim at theoretical description of the SD states in 32S and in four neighboring nuclei: 33S, 31S, 33Cl, and 31P. We present predictions pertaining to detailed spectroscopic information on excitation energies, spins, moments of inertia, and quadrupole moments of the SD rotational bands. All these observables may, in a very near future, become available within the discrete-spectroscopy measurements using large detector arrays; these observables have already been obtained experimentally in the other groups of SD nuclei.
The paper is organized as follows. After briefly presenting in Sec. 2 the theoretical methods we use in this study, in Sec. 3 we discuss the deformed-shell gaps and Coulomb effects in 32S, we present a classification of the SD bands, and describe the level crossings. Results of calculations for the SD bands in 32S are presented in Sec. 4, and those for 33S, 31S, 33Cl, and 31P in Sec. 5. Finally, in Sec. 6 we briefly discuss the question of the stability of SD bands, and Sec. 7 presents our conclusions.