  In this paper, we discuss an interaction between complex geometry and
integrable systems. Section 1 reviews the classical results on integrable
systems. New examples of integrable systems, which have been discovered, are
based on the Lax representation of the equations of motion. These systems can
be realized as straight line motions on a Jacobi variety of a so-called
spectral curve. In section 2, we study a Lie algebra theoretical method leading
to integrable systems and we apply the method to several problems. In section
3, we discuss the concept of the algebraic complete integrability (a.c.i.) of
hamiltonian systems. Algebraic integrability means that the system is
completely integrable in the sens of the phase space being folited by tori,
which in addition are real parts of a complex algebraic tori (abelian
varieties). The method is devoted to illustrate how to decide about the a.c.i.
of hamiltonian systems and is applied to some examples. Finally, in section 4
we study an a.c.i. in the generalized sense which appears as covering of a.c.i.
system. The manifold invariant by the complex flow is covering of abelian
variety.
