  We consider solutions of the non-linear von Neumann equation involving
Jacobi's elliptic functions sn, cn, and dn, and 3 linearly independent
operators. In two cases one can construct a state-dependent Hamiltonian which
is such that the corresponding non-linear von Neumann equation is solved by the
given density operator. We prove that in a certain context these two cases are
the only possibilities to obtain special solutions of this kind. Well-known
solutions of the reduced Maxwell-Bloch equations produce examples of each of
the two cases. Also known solutions of the non-linear von Neumann equation in
dimension 3 are reproduced by the present approach.
