  We demonstrate that the general model of a linearly time-dependent crossing
of two energy bands is integrable. Namely, the Hamiltonian of this model has a
quadratically time-dependent commuting operator. We apply this property to
four-state Landau-Zener (LZ) models that have previously been used to describe
the Landau-St\"uckelberg interferometry experiments with an electron shuttling
between two semiconductor quantum dots. The integrability then leads to simple
but nontrivial exact relations for the transition probabilities. In addition,
the integrability leads to a semiclassical theory that provides analytical
approximation for the transition probabilities in these models for all
parameter values. The results predict a dynamic phase transition, and show that
similarly-looking models belong to different topological classes.
