  We have investigated the processes N($\pi$, $\pi$)N and N($\pi$, $\eta$)N
close to eta threshold using a simple, nonrelativistic Lee model which has the
advantage of being analytically solvable. It is then possible to study the
Riemann sheets of the S-matrix and the behavior of its resonance poles
especially close to threshold. A theoretical simulation of the experimental
cusp effect at eta threshold leads to a characteristic distribution of poles on
the Riemann sheets. We find a pole located in the $4^{th}$ Riemann sheet that
up to now has not been discussed. It belongs to the cusp peak at eta threshold.
In addition we obtain the surprising result using the Lee model that the
resonance $S_{11}(1535)$ does not play a large role. The main features of the
experimental data can be reproduced without explicitly introducing this
resonance. Furthermore, we have also studied the reactions N($\gamma$, $\pi$)N
and N($\gamma$, $\eta$)N and find reasonable agreement between the data and
both models with and without the $S_{11}(1535)$ resonance.
