  A Darboux-type method of solving the nonlinear von Neumann equation $i\dot
\rho=[H,f(\rho)]$, with functions $f(\rho)$ commuting with $\rho$, is
developed. The technique is based on a representation of the nonlinear equation
by a compatibility condition for an overdetermined linear system. von Neumann
equations with various nonlinearities $f(\rho)$ are found to possess the
so-called self-scattering solutions. To illustrate the result we consider the
Hamiltonian $H$ of a one-dimensional harmonic oscillator and
$f(\rho)=\rho^q-2\rho^{q-1}$ with arbitary real $q$. It is shown that
self-scattering solutions possess the same asymptotics for all $q$ and that
different nonlinearities may lead to effectively indistinguishable evolutions.
The result may have implications for nonextensive statistics and experimental
tests of linearity of quantum mechanics.
