This paper is devoted to the investigation of the role of a single degree-of-freedom (d.o.f) nonlinear energy sink (NES) with nonlinear damping characteristics in annihilating undesired periodic response regimes and simultaneously preserving strongly modulated responses (SMR) in a harmonically excited three d.o.f system consisting of two linear coupled oscillators and an NES attached to it. The NES parameters are tuned such that they are able to eliminate all undesired periodic branches and to have the SMR region in the vicinity of both modes of the system concurrently. The tuning procedure leads to a maximum SMR interval which is achievable using this type of the NES in the vicinity of each excited mode. According to the results, design requirements are achieved using this type of the NES for a two d.o.f primary system for the first time. But, further increase in the d.o.f of the system will lead to a restriction of the range of harmonic excitation amplitude of the first oscillator for which the design requirements would be satisfied.