  We analytically construct the wave function that, for a given initial state,
produces a prescribed density for a quantum ring with two non-interacting
particles in a singlet state. In this case the initial state is completely
determined by the initial density, the initial time-derivative of the density
and a single integer that characterizes the (angular) momentum of the system.
We then give an exact analytic expression for the exchange-correlation
potential that relates two non-interacting systems with different initial
states. This is used to demonstrate how the Kohn-Sham procedure predicts the
density of a reference system without the need of solving the reference
system's Schr\"odinger equation. We further numerically construct the
exchange-correlation potential for an analytically solvable system of two
electrons on a quantum ring with a squared cosine two-body interaction. For the
same case we derive an explicit analytic expression for the
exchange-correlation kernel and analyze its frequency-dependence (memory) in
detail. We compare the result to simple adiabatic approximations and
investigate the single-pole approximation. These approximations fail to
describe the doubly-excited states, but perform well in describing the
singly-excited states.
