The nucleon pair shell model (NPSM) is cast into the so-called $M$ scheme for the cases with isospin symmetry and without isospin symmetry. The odd system and even system are treated on the same footing. The uncoupled commutators for nucleon pairs, which are suitable for the $M$ scheme, are given. Explicit formula of matrix elements in $M$ scheme for overlap, one-body operators, and two-body operators are obtained. It is found that the CPU time used in calculating the matrix elements in $M$ scheme is much shorter than that in the $J$ scheme of NPSM.

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.

The effects of possible noncollective pairs in even-even${}^{124\text{–}128}\mathrm{Sn}$ are studied in the nucleon-pair shell model, in which a few noncollective neutron pairs originating from the alignment of two neutrons in the $\nu h_{11/2}$ orbit are considered. From the low-lying-level energies, $B\left(E2\right)$ ratios, and excited states obtained from the model, it is shown that the yrast band structure can be explained as the evolution from vibrational to rotational type as a function of spin. The mechanism of the yrast band can be explained as band crossing between the ground-state band and the $S$ band constructed from the neutron alignment in the $\nu h_{11/2}$ orbit. The noncollective configurations may be crucial for describing the yrast states in even-even${}^{124,126,128}\mathrm{Sn}$ in the $SD$-pair shell model.