Rational version of Archimedes symplectomorphysm and birational Darboux coordinates on coadjoint orbit of $GL(N,C)$

A set of all linear transformations with a fixed Jordan structure $J$ is a symplectic manifold isomorphic to the coadjoint orbit $\mathcal{O}(J)$ of $GL(N,C)$. Any linear transformation may be projected along its eigenspace to (at least one) coordinate subspace of the complement dimension. The Jordan structure $\tilde{J}$ of the image is defined by the Jordan structure $J$ of the pre-image, consequently the projection $\mathcal O (J)\to \mathcal O (\tilde J)$isthemappingofthesymplecticmanifolds\mathrm{.}Itisprovedthatthefiber$\mathcal E$oftheprojectionisalinearsymplecticspaceandthemap$\mathcal O(J) \to \mathcal E \times \mathcal O (\tilde J)$ is a birational symplectomorphysm. The iteration of the procedure gives the isomorphism between $\mathcal{O}(J)$ and the linear symplectic space, which is the direct product of all the fibers of the projections. The Darboux coordinates on $\mathcal O(J)$ are pull-backs of the canonical coordinates on the linear spaces in question.