Nonlinear von Neumann-type equations: Darboux invariance and spectra

Generalized Euler-Arnold-von Neumann density matrix equations can be solved by a binary Darboux transformation given here in a new form: $\rho[1]=e^{P\ln(\mu/\nu)}\rho e^{-P\ln(\mu/\nu)}$ where $P=P^2$ is explicitly constructed in terms of conjugated Lax pairs, and $\mu$, $\nu$ are complex. As a result spectra of $\rho$ and $\rho[1]$ are identical. Transformations allowing to shift and rescale spectrum of a solution are introduced, and a class of stationary seed solutions is discussed.