  We consider a general theory of Landau-Zener transitions in a three-level
system. Based on a classification of three level crossings we express the
Landau - Zener Hamiltonians in terms of two bases: i) spin S=1 SU(2) operators
and ii) SU(3) Gell - Mann matrices. We show that the generic Hamiltonians being
non-linear in terms of the SU(2) group generators become linear in the SU(3)
basis. If the diabatic states of the SU(3) Landau - Zener Hamiltonian form a
triangle, the interference between two paths results in formation of "beats"
and "steps" pattern in the time-dependent transition probability. The
characteristic time scales describing the "beats" and "steps" depend on a dwell
time through the triangle. These scales are related to the geometric size of
the interferometer. We formulate the SU(3) Landau - Zener problem in terms of
Bloch dynamics of a unit vector and find a solution of eight-dimensional Bloch
equations in the limit of a non-adiabatic transition. Possible experiments in
triangular and linearly arranged triple quantum dots where Landau - Zener
interferometry can be used for finding manifestations of SU(3) symmetry are
discussed.
