  We study the slow time evolution of localized states of the open-boundary
Lieb lattice when a magnetic flux is applied perpendicularly to the lattice and
increased linearly in time. In this system, Dirac cones periodically disappear,
reappear and touch the flat band as the flux increases. We show that the slow
time evolution of a localized state in this system is analogous to that of a
zero-energy state in a three-level system whose energy levels intersect
periodically and that this evolution can be mapped into a classical precession
motion with a precession axis that rotates as times evolves. Beginning with a
localized state of the Lieb lattice, as the magnetic flux is increased linearly
and slowly, the evolving state precesses around a state with a small itinerant
component and the amplitude of its localized component oscillates around a
constant value (below but close to 1), except at multiples of the flux quantum
where it may vary sharply. This behavior reflects the existence of an electric
field (generated by the time-dependent magnetic field) which breaks the C4
symmetry of the constant flux Hamiltonian.
