  Exactly solvable multistate Landau-Zener (MLZ) models are associated with
families of operators that commute with the MLZ Hamiltonians and depend on time
linearly. There can also be operators that satisfy the integrability conditions
with the MLZ Hamiltonians but depend on time quadratically. We show that, among
the MLZ systems, such time-quadratic operators are much more common. We
demonstrate then that such operators generally lead to constraints on the
independent variables that parametrize the scattering matrix. We show how such
constraints lead to asymptotically exact expressions for the transition
probabilities in the adiabatic limit of a three-level MLZ model. New fully
solvable MLZ systems are also found.
