  The purpose of this paper is to construct a generalized r-matrix structure of
finite dimensional systems and an approach to obtain the algebro-geometric
solutions of integrable nonlinear evolution equations (NLEEs). Our starting
point is a generalized Lax matrix instead of usual Lax pair. The generalized
r-matrix structure and Hamiltonian functions are presented on the basis of
fundamental Poisson bracket. It can be clearly seen that various nonlinear
constrained (c-) and restricted (r-) systems, such as the c-AKNS, c-MKdV,
c-Toda, r-Toda, c-Levi, etc, are derived from the reduction of this structure.
All these nonlinear systems have {\it r}-matrices, and are completely
integrable in Liouville's sense. Furthermore, our generalized structure is
developed to become an approach to obtain the algebro-geometric solutions of
integrable NLEEs. Finally, the two typical examples are considered to
illustrate this approach: the infinite or periodic Toda lattice equation and
the AKNS equation with the condition of decay at infinity or periodic boundary.
