  We show several examples of integrable systems related to special K3 and
rational surfaces (e.g., an elliptic K3 surface, a K3 surface given by a double
covering of the projective plane, a rational elliptic surface, etc.). The
construction, based on Beauvilles's general idea, is considerably simplified by
the fact that all examples are described by hyperelliptic curves and Jacobians.
This also enables to compare these integrable systems with more classical
integrable systems, such as the Neumann system and the periodic Toda chain,
which are also associated with rational surfaces. A delicate difference between
the cases of K3 and of rational surfaces is pointed out therein.
