  The concept of quantum integrability has been introduced recently for quantum
systems with explicitly time-dependent Hamiltonians. Within the multistate
Landau-Zener (MLZ) theory, however, there has been a successful alternative
approach to identify and solve complex time-dependent models. Here we compare
both methods by applying them to a new class of exactly solvable MLZ models.
This class contains systems with an arbitrary number $N\ge 4$ of interacting
states and shows a quickly growing with $N$ number of exact adiabatic energy
crossing points, which appear at different moments of time. At each $N$,
transition probabilities in these systems can be found analytically and exactly
but complexity and variety of solutions in this class also grow with $N$
quickly. We illustrate how common features of solvable MLZ systems appear from
quantum integrability and develop an approach to further classification of
solvable MLZ problems.
