The Anderson-Rickayzen equations of motion for a superconductor derived within the random-phase approximation (RPA) are used to investigate the collective excitations of superconductors. A spherical harmonic expansion is made of the two-body interaction potential V(k, k') and a spectrum of excitations whose energies lie within the energy gap 2∆ is obtained. These excitations may be characterized by the quantum numbers L and M involved in the potential expansion. For an L-state exciton to exist, the L-wave part of the potential must be attractive at the Fermi surface. Odd-L excitons have unit spin and may be considered as spin waves. For s-state pairing in the superconducting ground state, the plasmon mode corresponds to the L=0 exciton whose energy is strongly modified by the long-range Coulomb interaction. For a general potential several bound states may exist for given L and M. If the L-wave potential is stronger than the s-wave part of the potential, the system is unstable with respect to formation of L-state excitons. In this case, the ground state is formed with L-state pairing, special cases of which are the p-state pairing considered by Fisher and the d-state pairing proposed recently by several authors for the ground state of He3 and nuclear matter. Corrections to the Anderson-Rickayzen equations are discussed which lead to a new set of exciton states if the L-wave potential is repulsive. These excitons are interpreted as bound electron-hole pairs, as opposed to the particle-particle excitons present with an attractive L-wave potential.