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• Novel ginkgo-leaf spectrum is reported in cold atomic systems. • Triple degeneracy points and degeneracy line of nonlinear spectrum are predicted. • That the band curvature and current density can be controlled by interaction is proved. Band structure is an important tool to characterize the physical properties of periodic systems. In this work, we investigate the nonlinear spectrum and current density in cold atoms with interaction-dependent dispersion (IDD). The results reveal that IDD causes the nonlinear loop structure, which means that the dynamical stability is destroyed, resulting in the loss of superfluidity. More interestingly, the repulsive IDD induces a more complex deformation of spectrum. Different from the previous studies, the conversion between positive and negative curvature loop structures appears in the system. Furthermore, there will be a pair of triple degeneracy points on the lower band (n = 2 n c) and a double degeneracy line between them, which will uncover new physical properties, e.g., nonlinear Landau-Zener tunneling. Since the nonlinear interaction of cold atoms is highly controllable through Feshbach resonance, this fascinating phenomenon is expected to be realized in cold atomic systems in the near future. [ABSTRACT FROM AUTHOR]