  We solve the non-stationary Schrodinger equation for several time-dependent
Hamiltonians, such as the BCS Hamiltonian with an interaction strength
inversely proportional to time, periodically driven BCS and linearly driven
inhomogeneous Dicke models as well as various multi-level Landau-Zener
tunneling models. The latter are Demkov-Osherov, bow-tie, and generalized
bow-tie models. We show that these Landau-Zener problems and their certain
interacting many-body generalizations map to Gaudin magnets in a magnetic
field. Moreover, we demonstrate that the time-dependent Schrodinger equation
for the above models has a similar structure and is integrable with a similar
technique as Knizhnikov-Zamolodchikov equations. We also discuss applications
of our results to the problem of molecular production in an atomic Fermi gas
swept through a Feshbach resonance and to the evaluation of the Landau-Zener
transition probabilities.
