This work studies a problem of passive mitigation by means of one nonlinear energy sink (NES) of a dynamic instability arising from two unstable modes of a primary structure. For that, a phenomenological friction model with four-degrees-of-freedom (DOF) having two unstable modes (the primary structure) is coupled to a NES. We investigate analytically, using an asymptotic analysis, the response regimes resulting from the interaction between the two unstable linear modes of the primary structure and the NES. The analytical results are compared and complemented by direct numerical simulations of the coupled system. In each situation, a good agreement is observed which allows us to validate the presented asymptotic analysis.