Birational Darboux Coordinates on Nilpotent Coadjoint Orbits of Classical Complex Lie Groups, the Case of 2 × 2 Jordan Blocks

We consider the problem of constructing birational Darboux coordinates on nilpotent coadjoint orbits of the complex Lie groups SO(N, ℂ) and Sp(N, ℂ). The nilpotent case is the most difficult one. Difficulties arise if the Jordan form of matrices from the orbit under consideration contains Jordan blocks of sizes of different parity. The desired coordinates have been found on orbits consisting of matrices with 1 × 1 and 2 × 2 Jordan blocks. Explicit formulas for them are given in the paper.