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Recently the interacting bosom model with $SU(3)$ higher-order interactionswas proposed by one of the authors, wherein unexpected $\gamma$-softness can beemerged in this new model. This stimulates further discussion on theconnections of the new $\gamma$-softness and the realistic $\gamma$-softnuclei. In this paper, $E(5)$-like $\gamma$-softness arises when the $SU(3)$fourth-order interaction $\hat{C}_{2}^{2}[SU(3)]$ is considered. And thecorresponding transitional behaviors are similar to that from the $U(5)$ limitto the $O(6)$ limit in previous IBM-1, which provides a novel perspective forunderstanding the new model. Low-lying spectra, $B(E2)$ values between thelow-lying states and quadrupole moment of the first $2_{1}^{+}$ state areinvestigated. Experimental data of the $E(5)$-like nucleus $^{82}$Kr arecompared with the calculated theoretical results, where the calculatedlow-lying level energies and the associated $B(E2)$ values fit very well withthe experimental data.

$E(5)$-like emerging $\gamma$-softness in $^{82}$Kr