  This paper investigates the fast Hamiltonian control of $n=2$ density
operators by continuously varying the flag as one moves away from the
completely mixed state. In general, the critical points and zeros of the purity
derivative can only be solved analytically in the limit of minimal purity. We
derive differential equations that maintain these features as the purity
increases. In particular, there is a thread of points in the Bloch ball that
locally maximizes the purity derivative, and a corresponding thread that
minimizes it. Additionally, we show there is a closed surface of points inside
of which the purity derivative is positive, and inside of which is negative. We
argue that this approach may be useful in studying higher-dimensional systems.
