  Mixed-moment models, introduced before for one space dimension, are a
modification of the method of moments applied to a (linear) kinetic equation,
by choosing mixtures of different partial moments. They are well-suited to
handle such equations where collisions of particles are modelled with a
Laplace-Beltrami operator. We generalize the concept of mixed moments to two
dimension. The resulting hyperbolic system of equations has desirable
properties, removing some drawbacks of the well-known $\MN[1]$ model. We
furthermore provide a realizability theory for a first-order system of mixed
moments by linking it to the corresponding quarter-moment theory. Additionally,
we derive a type of Kershaw closures for mixed- and quarter-moment models,
giving an efficient closure (compared to minimum-entropy models). The derived
closures are investigated for different benchmark problems.
