We present an analytical study of the conservative and dissipative dynamics of a two-degree-of-freedom (DOF) system consisting of a linear oscillator coupled to a bistable light attachment. The main objective of the paper is to study the beneficial effect of the bistability on passive nonlinear targeted energy transfer from the impulsively excited linear oscillator to an appropriately designed attachment. As a numerical study of the problem has shown in a companion paper (Romeo, F., Sigalov, G., Bergman, L. A., and Vakakis, A. F., 2013, “Dynamics of a Linear Oscillator Coupled to a Bistable Light Attachment: Numerical Study,” J. Comput. Nonlinear Dyn. (submitted)) there is an essential difference in the system's behavior when compared to the conventional case of a monostable attachment. On the other hand, some similarity to the behavior of an oscillator with rotator attachment has been revealed. It relates, in particular, to the generation of nonconventional nonlinear normal modes and to the existence of two qualitatively different types of dynamics. We find that all numerical results can be explained in the framework of fundamental (1:1) and superharmonic (1:3) resonances (for large energies), as well as a subharmonic resonance (for small energies). This allows us to use the concept of limiting phase trajectories (LPTs) introduced earlier by one of the authors, and to derive accurate analytical approximations to the dynamics of the problem in terms of nonsmooth generating functions.