Birational Darboux Coordinates on (Co)Adjoint Orbits of $\operatorname{GL}(N,\mathbb C)$

The set of all linear transformations with a fixed Jordan structure $\mathcal J$ is a symplectic manifold isomorphic to the coadjoint orbit $\mathcal O (\mathcal J)$ of the general linear group $\operatorname{GL}(N,{\mathbb C})$. Any linear transformation can be projected along its eigenspace onto a coordinate subspace of complementary dimension. The Jordan structure $\tilde{\mathcal J}$ of the image under the projection is determined by the Jordan structure $\mathcal J$ of the preimage; consequently, the projection $\mathcal O (\mathcal J)\to \mathcal O (\tilde{\mathcal J})$ is a mapping of symplectic manifolds.
It is proved that the fiber $\mathscr{E}$ of the projection is a linear symplectic space and the map $\mathcal O(\mathcal J) \stackrel{\sim}{\to} \mathscr{E} \times \mathcal O (\tilde{\mathcal J})$ is a birational symplectomorphism. Successively projecting the resulting transformations along eigensubspaces yields an isomorphism between $\mathcal O (\mathcal J)$ and the linear symplectic space being the direct product of all fibers of the projections. The Darboux coordinates on $\mathcal O(\mathcal J)$ are pullbacks of the canonical coordinates on this linear symplectic space.
Canonical coordinates on orbits corresponding to various Jordan structures are constructed as examples.