  In this series of papers we present a detailed study of the
particle--particle collective excitations of the Hubbard model, and their
contribution to the density and spin excitation spectrum. In the first paper,
we shall investigate the singlet particle--particle pair with momentum
$(\pi,\pi)$, the $\eta$ particle, of the negative--$U$ Hubbard model. We review
three previously obtained theorems about the $\eta$ particle and develop a
self-consistent linear response theory which takes into account its
contribution to the density excitation spectrum in the superconducting state.
We show that this self--consistent theory agrees with the exact theorems as
well as the results of numerical Monte Carlo simulations.
