I will present here some examples of integrable systems, all of them defined on the moduli space of flat connections on a trivial bundle over a surface. These examples have been constructed by (loldman, Jeffrey and Weitsman. lock. Alekseev, so that there will be nothing new in these notes. However, it seems to me that a general presentation is lacking in the literature, and that this i.s a pity, as so many beautiful ideas are involved.
The reasons why one is willing to consider such things are various and the motivations really depend on the authors. My two starting points will be:
1. It is interesting to understand the geometry of the moduli space. There are a lot of motivations (some of them coming from physics).
2. In order to understand the geometry of a Poisson manifold, it is very helpful
to have an integrable system on it.