  It is shown how Darboux coordinates on a reduced symplectic vector space may
be used to parametrize the phase space on which the finite gap solutions of
matrix nonlinear Schr\"odinger equations are realized as isospectral
Hamiltonian flows. The parametrization follows from a moment map embedding of
the symplectic vector space, reduced by suitable group actions, into the dual
$\tilde\grg^{+*}$ of the algebra $\tilde\grg^+$ of positive frequency loops in
a Lie algebra $\grg$. The resulting phase space is identified with a Poisson
subspace of $\tilde\grg^{+*}$ consisting of elements that are rational in the
loop parameter. Reduced coordinates associated to the various Hermitian
symmetric Lie algebras $(\grg,\grk)$ corresponding to the classical Lie
algebras are obtained.
