Geodesic
On rational symplectic parametrization of the coadjoint orbit of $\mathrm{GL}(N)$. Diagonalizable case
Algebra i analiz, Tome 22 (2010) no. 3, pp. 16-31.

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A method for constructing birational Darboux coordinates on a coadjoint orbit of the general linear group is presented. This method is based on the Gauss decomposition of a matrix in the product of an upper-triangular and a lower-triangular matrix. The method works uniformly for the orbits formed by the diagonalizable matrices of any size and for arbitrary dimensions of the eigenspaces.
Keywords: Darboux coordinates, symplectic form, Poisson bracket, coadjoint orbit. 
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     title = {On rational symplectic parametrization of the coadjoint orbit of $\mathrm{GL}(N)$. {Diagonalizable} case},
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}
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M. V. Babich; S. E. Derkachov. On rational symplectic parametrization of the coadjoint orbit of $\mathrm{GL}(N)$. Diagonalizable case. Algebra i analiz, Tome 22 (2010) no. 3, pp. 16-31. http://geodesic.mathdoc.fr/item/AA_2010_22_3_a1/

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