  Darboux coordinates are constructed on rational coadjoint orbits of the
positive frequency part $\wt{\frak{g}}^+$ of loop algebras. These are given by
the values of the spectral parameters at the divisors corresponding to
eigenvector line bundles over the associated spectral curves, defined within a
given matrix representation. A Liouville generating function is obtained in
completely separated form and shown, through the Liouville-Arnold integration
method, to lead to the Abel map linearization of all Hamiltonian flows induced
by the spectral invariants. Serre duality is used to define a natural
symplectic structure on the space of line bundles of suitable degree over a
permissible class of spectral curves, and this is shown to be equivalent to the
Kostant-Kirillov symplectic structure on rational coadjoint orbits. The general
construction is given for $\frak{g}=\frak{gl}(r)$ or $\frak{sl}(r)$, with
reductions to orbits of subalgebras determined as invariant fixed point sets
under involutive automorphisms. The case $\frak{g=sl}(2)$ is shown to reproduce
the classical integration methods for finite dimensional systems defined on
quadrics, as well as the quasi-periodic solutions of the cubically nonlinear
Schr\"odinger equation. For $\frak{g=sl}(3)$, the method is applied to the
computation of quasi-periodic solutions of the two component coupled nonlinear
Schr\"odinger equation.
