  We analyze Hamiltonians linear in the time variable for which the multistate
Landau-Zener problem is known to have an exact solution. We show that they
either belong to families of mutually commuting Hamiltonians polynomial in time
or reduce to the 2 x 2 Landau-Zener problem, which is considered trivially
integrable. The former category includes the equal slope, bow-tie, and
generalized bow-tie models. For each of these models we explicitly construct
the corresponding families of commuting matrices. The equal slope model is a
member of an integrable family that consists of the maximum possible number
(for a given matrix size) of commuting matrices linear in time. The bow-tie
model belongs to a previously unknown, similarly maximal family of quadratic
commuting matrices. We thus conjecture that quantum integrability understood as
the existence of nontrivial parameter-dependent commuting partners is a
necessary condition for the Landau-Zener solvability. Descendants of the 2 x 2
Landau-Zener Hamiltonian are e.g. general SU(2) and SU(1,1) Hamiltonians,
time-dependent linear chain, linear, nonlinear, and double oscillators. We
explicitly obtain solutions to all these Landau-Zener problems from the 2 x 2
case.
