  We investigate the superconducting (SC), charge-density wave (CDW), and
antiferromagnetic (AFM) phases in the extended Hubbard model at zero
temperature and half-filling. We employ the iterated equations of motion
approach to compute the two-particle Green's functions and their spectral
densities. This renders a comprehensive analysis of the behavior of collective
excitations possible as the model's parameters are tuned across phase
transitions. We identify the well-known amplitude (Higgs) and phase
(Anderson-Bogoliubov) modes within the superconducting phase and observe a
similar excitation (cooperon) in the CDW phase which shifts towards zero energy
as the system approaches the phase transition to the SC phase. In the CDW
phase, close to the phase transition to the AFM phase, we find a collective
mode, an exciton, that does not change significantly and another mode, a
longitudinal magnon, that emerges from the two-particle continuum as the system
approaches the phase transition to the AFM phase. It becomes identical with the
former at the transition. In the AFM phase, their roles are reversed.
Additionally, we find a transversal Goldstone magnon located at zero energy.
