  We compute Landau-Zener probabilities for 3-level systems with a linear sweep
of the uncoupled energy levels of the 3$\times$3 Hamiltonian $H(t)$. Two
symmetry classes of Hamiltonians are studied: For $H(t) \in$ su(2) (expressible
as a linear combination of the three spin 1 matrices), an analytic solution to
the problem is obtained in terms of the parabolic cylinder $D$ functions. For
$H(t) \in$ su(3) (expressible as a linear combination of the eight Gell-Mann
matrices), numerical solutions are obtained. In the adiabatic regime, full
population transfer is obtained asymptotically at large time, but at
intermediate times, all three levels are populated and St\"uckelberg
oscillations are typically manifest. For the open system, (wherein interaction
with a reservoir occurs), we numerically solve a Markovian quantum master
equation for the density matrix with Lindblad operators that models interaction
with isotropic white Gaussian noise. We find that St\"uckelberg oscillations
are suppressed and that the temporal decay law of the population probabilities
is not a simple exponential.
