  This paper is devoted to the study of some connections between coadjoint
orbits in infinite dimensional Lie algebras, isospectral deformations and
linearization of dynamical systems. We explain how results from deformation
theory, cohomology groups and algebraic geometry can be used to obtain insight
into the dynamics of integrable systems. Another part will be dedicated to the
study of infinite continued fraction, orthogonal polynomials, the isospectral
deformation of periodic Jacobi matrices and general difference operators from
an algebraic geometrical point of view. Some connections with Cauchy-Stieltjes
transform of a suitable measure and Abelian integrals are given. Finally the
notion of algebraically completely integrable systems is explained, techniques
to solve such systems are presented and some interesting cases appear as
coverings of such dynamical systems. These results are exemplified by several
problems of dynamical systems of relevance in mathematical physics.
