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Prolate-oblate shape phase transition is an interesting topic in nuclearstructure, which is useful for understanding the intrinsic interactions betweennucleons. Recently, the interacting boson model with $SU(3)$ higher-orderinteractions was proposed, in which the prolate shape and the oblate shape arenot described in a mirror symmetric way. This asymmetric description seems morerealistic. The level evolutions, $B(E2)$ values and other important indicatorsshowing the prolate-oblate asymmetric transitions are investigated in detail,and realistic structure evolutions from $^{180}$Hf to $^{200}$Hg are compared.A key finding is that, the average deformation of the prolate shape is nearlytwice the one of the oblate shape. These results, together with the successfuldescription of the $B(E2)$ anomaly in $^{168,170}$Os, $^{172}$Pt, the$\gamma$-soft properties of $^{196}$Pt, $^{82}$Kr and the normal states of$^{110}$Cd, support the validity of the new model.

Prolate-oblate asymmetric shape phase transition in the interacting boson model with SU (3) higher-order interactions