  The aim of these lectures is to show that the methods of classical
Hamiltonian mechanics can be profitably used to solve certain classes of
nonlinear partial differential equations. The prototype of these equations is
the well-known Korteweg-de Vries (KdV) equation. In these lectures we touch the
following subjects: i) The birth and the role of the method of Poisson pairs
inside the theory of the KdV equation; ii) the theoretical basis of the method
of Poisson pairs; iii) the Gel'fand-Zakharevich theory of integrable systems on
bihamiltonian manifolds; iv) the Hamiltonian interpretation of the Sato picture
of the KdV flows and of its linearization on an infinite-dimensional
Grassmannian manifold. v)the reduction technique and its use to construct
classes of solutions; iv) the role of the technique of separation of variables
in the study of the reduced systems. vii) some relations intertwining the
method of Poisson pairs with the method of Lax pairs.

Exactly Solvable and Integrable Systems