  A method of constructing of Darboux coordinates on a space that is a
symplectic reduction with respect to a diagonal action of $GL(m})$ on a
Cartesian product of $N$ orbits of coadjoint representation of $GL(m)$ is
presented. The method gives an explicit symplectic birational isomorphism
between the reduced space on the one hand and a Cartesian product of $N-3$
coadjoint orbits of dimension $m(m-1)$ on an orbit of dimension $(m-1)(m-2)$ on
the other hand. In a generic case of the diagonalizable matrices it gives just
the isomorphism that is birational and symplectic between some open, in a
Zariski topology, domain of the reduced space and the Cartesian product of the
orbits in question. The method is based on a Gauss decomposition of a matrix on
a product of upper-triangular, lower-triangular and diagonal matrices. It works
uniformly for the orbits formed by diagonalizable or not-diagonalizable
matrices. It is elaborated for the orbits of maximal dimension that is
$m(m-1)$.
