  We show that the solution of a multistate system composed of N degenerate
lower (ground) states and one upper (excited) state can be reduced by using the
Morris-Shore transformation to the solution of a two-state system involving
only the excited state and a (bright) superposition of ground states. In
addition, there are N-1 dark states composed of ground states. We use this
decomposition to derive analytical solutions for degenerate extensions of the
most popular exactly soluble models: the resonance solution, the Rabi,
Landau-Zener, Rosen-Zener, Allen-Eberly and Demkov-Kunike models. We suggest
various applications of the multistate solutions, for example, as tools for
creating multistate coherent superpositions by generalized resonant pi-pulses.
We show that such generalized pi-pulses can occur even when the upper state is
far off resonance, at special detunings, which makes it possible to operate in
the degenerate ground-state manifold without populating the (possibly lossy)
upper state, even transiently.
