This paper investigates the different response regimes of a cutting tool on a lathe strongly coupled to a nonlinear energy sink (NES). The system consists of a linear oscillator representing a flexible lathe tool, subject to a regenerative cutting force and strongly coupled to a NES. Only the linear part of the cutting force is considered. The equations of motion are analyzed via the method of multiple scales. Condition of elimination of secular terms permit to derive equation of the slow invariant manifold (SIM) and the behavior of the system has been explained by studying the location of the fixed points of the slow flow on the SIM. Different types of responses are revealed such as periodic response and also strongly modulated response (SMR) that are not related to the fixed points of the slow flow. Analytic results are then compared with numerical simulations. The potential benefit of the NES to control machining chatter is demonstrated. It should be interesting to include a non-linear cutting law, and to study the system behavior for a larger set of parameters and an optimized NES.