  We analyze a class of coupled quantum systems whose dynamics can be
understood via two uncoupled, lower-dimensional quantum settings with auxiliary
interactions. The general reduction scheme, based on algebraic properties of
the potential term, is discussed in detail for two-dimensional Dirac
Hamiltonian. We discuss its possible application in description of Dirac
fermions in graphene or bilayer graphene in presence of distortion scattering
or spin-orbit interaction. The framework is illustrated on the explicit
examples where the interaction depends on two spatial coordinates or it is
time-dependent.
