Birational Darboux coordinates on nilpotent coadjoint orbits classical complex Lie groups, Jordan blocks $2\times2$

A problem of the constructing of the birational Darboux coordinates on the nilpotent coadjoint orbits of the complex Lie groups $SO(N,\mathbb C)$ and $Sp(N,\mathbb C)$ is considered. The nilpotent case is the most difficult case of the orbits. The difficulties arise if the Jordan blocks of the different parities of the sizes present in the Jordan form of the matrices from the orbit. The desired coordinates has been found on the orbits consisting of the matrices with the Jordan blocks of the sizes one and two. The explicit formulae for the coordinates are presented.