<p>Vari-potential bi-stable nonlinear energy sink (VBNES) has broadband and high-efficiency vibration absorption performance. This study presents the first comprehensive analytical investigation on the periodic and quasi-periodic dynamics of a system coupling a VBNES with a linear oscillator (LO). Firstly, the exact and approximate theoretical models of the LO-VBNES are established based on Lagrange equation and Taylor series expansion. The potential energy and restoring force surfaces are predicted, and vari-potential barrier effect (VPBE) is verified by numerical response trajectory tracking on these surfaces. Secondly, the complexification-averaging (CX-A) and multiple scales method (MSM) are used to derive the complex-averaged modulation equations of the LO-VBNES. The slow invariant manifold (SIM) of the LO-VBNES is predicted by detecting the fast average system, and the periodic and non-periodic orbits are validated with numerical results. The influences of system parameters on the SIM and the fold singularities of the SIM are discussed. Furthermore, the amplitude-frequency curves of the periodic responses in the 1:1 resonance region are investigated via the slow average system, which has single and multiple responses separated by saddle-node (SN) bifurcations. The analytical and numerical results indicate that the multi-responses are low- and high-energy harmonics, chaotic, or strongly modulated responses (SMR). The influences of different parameters on the amplitude-frequency response and vibration suppression performance are analyzed. Finally, the analytical expressions of the excitation thresholds of the SMR are predicted, and the numerical verification shows that the analytical thresholds are effective.</p>