  Recently the interacting bosom model with $SU(3)$ higher-order interactions
was proposed by one of the authors, wherein unexpected $\gamma$-softness can be
emerged in this new model. This stimulates further discussion on the
connections of the new $\gamma$-softness and the realistic $\gamma$-soft
nuclei. 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 the
corresponding transitional behaviors are similar to that from the $U(5)$ limit
to the $O(6)$ limit in previous IBM-1, which provides a novel perspective for
understanding the new model. Low-lying spectra, $B(E2)$ values between the
low-lying states and quadrupole moment of the first $2_{1}^{+}$ state are
investigated. Experimental data of the $E(5)$-like nucleus $^{82}$Kr are
compared with the calculated theoretical results, where the calculated
low-lying level energies and the associated $B(E2)$ values fit very well with
the experimental data.
