  We have studied a generalized three band crossing model in 2D, the
generalized $\alpha - T_3$ lattice, ranging from the pseudospin-1 Dirac
equation through a quadratic+flat band touching to the pseudospin-1/2 Dirac
equation. A general method is presented to determine the operator form of the
Green's function, being gauge and representation independent. This yields the
Landau level structure in a quantizing magnetic field and the longitudinal and
transversal magneto-optical conductivities of the underlying system Although
the magneto-optical selection rules allow for many transitions between Landau
levels, the dominant one stems from exciting a particle from/to the flat band
to/from a propagating band. The Hall conductivity from each valley is rational
(not quantized at all), in agreement with Berry phase considerations, though
their sum is always integer quantized.
