Prolate-oblate shape phase transition is an interesting topic in nuclear structure, which is useful for understanding the intrinsic interactions between nucleons. Recently, the interacting boson model with SU(3) higher-order interactions was proposed, in which the prolate shape and the oblate shape are not described in a mirror symmetric way. This asymmetric description seems more realistic. The level evolutions, $B\left(E2\right)$ values, and other important indicators showing the prolate-oblate asymmetric transitions are investigated in detail, and realistic structure evolutions from ${}^{180}\mathrm{Hf}$ to ${}^{200}\mathrm{Hg}$ are compared. A key finding is that the average deformation of the prolate shape is nearly twice the one of the oblate shape. These results, together with the successful description of the $B\left(E2\right)$ anomaly in ${}^{168,170}\mathrm{Os}, {}^{172}\mathrm{Pt}$, the $\gamma$-soft properties of ${}^{196}\mathrm{Pt}, {}^{82}\mathrm{Kr}$, and the normal states of ${}^{110}\mathrm{Cd}$, support the validity of the new model.

Following the construction approach to successfully describe the $B\left(E2\right)$ anomaly in the interacting boson model with SU(3) higher-order interactions, the O(6) higher-order interactions are also considered to be used to investigate this anomaly, but the numerical calculations show that this construction approach is unphysical. This implies that O(6) symmetry is insufficient to construct a general theory to describe the collective properties in nuclei.

Recently the interacting boson model with SU(3) higher-order interactions was proposed by one of the authors, in which new $\gamma$ softness can emerge. This stimulates further discussion on the connections between the new $\gamma$ softness and realistic $\gamma$-soft nuclei. In this paper, E(5)-like $\gamma$ softness arises when the SU(3) fourth-order interaction ${\widehat{C}}_2^2\left[\text{SU(3)}\right]$ is considered. The corresponding transitional behaviors are similar to those from the U(5) limit to the O(6) limit in the previous interacting boson model, IBM-1, which provides a novel perspective for understanding the new model. Low-lying spectra, $B\left(E2\right)$ values, and quadrupole moment of the first $2_1^{+}$ state are investigated. The E(5)-like nucleus ${}^{82}\mathrm{Kr}$ is exemplified, where the calculated low-lying level energies and the associated $B\left(E2\right)$ values fit very well with the experimental data.

Recently, studies have argued that a spherical-like spectrum emerges in the SU3-IBM, thus creating new approaches to understanding γ-softness in realistic nuclei. In a previous study, γ-softness with degeneracy of the ground and quasi-γ bands was observed. In this paper, another special point connected to the middle degenerate point is discussed. It is found to be related to the properties of ¹⁹⁶Pt. This emergent γ-softness is also shown to be important for understanding the prolate-oblate asymmetric shape phase transition. The low-lying spectra, values, and quadrupole moments in ¹⁹⁶Pt are discussed, and we show that the new model can account for several observed features. This is the first part of the discussion on the γ-soft-like spectrum of ¹⁹⁶Pt.

Dynamical symmetry plays a dominant role in the interacting boson model in elucidating nuclear structure, for which group theoretical or algebraic techniques are powerful. In this work, the higher-order interactions required in describing triaxial deformation in the interacting boson model are introduced to improve the fitting results to low-lying level energies, $B(E2)$ values and electric quadrupole moments of even–even nuclei. As an example of the model application, the low-lying excitation spectra and the electromagnetic transitional properties of even–even ^176−198Pt are fitted and compared to the experimental data and the results of the consistent-Q formalism. It is shown that the results obtained from the model are better than those of the original consistent-Q formalism, indicating the importance of the higher-order interactions in describing the structure and the shape phase evolution of these nuclei.