The underlying nonlinear dynamical behavior of a linear oscillator (LO) attached with nonlinear energy sink (NES) is usually revealed by studying the LO-NES system on the frequency-energy plot (FEP). Accordingly, different kinds of periodic motions on the backbone curves of the nonlinear normal modes (NNMs) and their associated subharmonic branches are analyzed. Here, the considered unsymmetrical nonlinear energy sink (UNES) incorporates a cubic stiffness element in one direction of the motion and a weak linear restoring coupling stiffness in both directions of the motion. This UNES is attached with the LO in the LO-UNES system which is studied here on the FEP. Therefore, the obtained FEP of the LO-UNES system is generated and analyzed. Accordingly, several unsymmetrical backbone curves of periodic oscillations between the UNES mass and the LO mass at 1:1 resonance have been obtained for broad range of nonlinear frequency levels.