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A set of all linear transformations with a fixed Jordan structure $J$ is asymplectic 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 Jordanstructure $\tilde J$ of the image is defined by the Jordan structure $J$ of thepre-image, consequently the projection $\mathcal O (J)\to \mathcal O (\tildeJ)$ is the mapping of the symplectic manifolds. It is proved that the fiber$\mathcal E$ of the projection is a linear symplectic space and the map$\mathcal O(J) \to \mathcal E \times \mathcal O (\tilde J)$ is a birationalsymplectomorphysm. The iteration of the procedure gives the isomorphism between$\mathcal O (J)$ and the linear symplectic space, which is the direct productof all the fibers of the projections. The Darboux coordinates on $\mathcalO(J)$ are pull-backs of the canonical coordinates on the linear spaces inquestion.

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