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A method of constructing of Darboux coordinates on a space that is asymplectic reduction with respect to a diagonal action of $GL(m})$ on aCartesian product of $N$ orbits of coadjoint representation of $GL(m)$ ispresented. The method gives an explicit symplectic birational isomorphismbetween 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 $(m-1)(m-2)$ onthe other hand. In a generic case of the diagonalizable matrices it gives justthe isomorphism that is birational and symplectic between some open, in aZariski topology, domain of the reduced space and the Cartesian product of theorbits in question. The method is based on a Gauss decomposition of a matrix ona product of upper-triangular, lower-triangular and diagonal matrices. It worksuniformly for the orbits formed by diagonalizable or not-diagonalizablematrices. It is elaborated for the orbits of maximal dimension that is$m(m-1)$.

Rational symplectic coordinates on the space of Fuchs equations $m \times m$-case