  Using the Dyson equation of motion for both the diagonal one-particle Green
function, $G(\vec{k},\omega)$ and off--diagonal Green function,
$F(\vec{k},\omega)$, at the level of the Hubbard--I decoupling scheme, we have
found that they have four poles symmetric in pairs, justifying a more
elaborated calculation done by the Z\"urich group by means of the $T$-Matrix
approach (Pedersen et al, Z. Physik B {\bf 103}, 21 (1997)) and the moment
approach of Nolting (Z. Physik {\bf 255}, 25 (1972)). We find that the energy
spectra and the weights of $G(\vec{k},\omega)$ and $F(\vec{k},\omega)$ have to
be calculated self-consistently. $G(\vec{k},\omega)$ satisfies the first two
moments while $F(\vec{k},\omega)$ the first sum rule. Our {\it order parameter}
$\alpha(T)$ is given by $1/N_s \sum_{{\vec{k}}}
\epsilon({\vec{k}})\Delta({\vec{k}})$. Due to the fact that we have a purely
local attractive interaction $\Delta(\vec{k})$ can be of {\it any} s--type
wave. However, for a {\it pure s--wave}, for which $\alpha(T) = 0$, we go back
to the mean--field $BCS$ results, with a renormalized chemical potential. In
this case, the off--diagonal Green function, $F(\vec{k},\omega)$, satisfies the
first two off--diagonal sum rules. We explicitly state the range of validity of
our approximation.
