  The system leading to phase segregation in two-component Bose-Einstein
condensates can be generalized to hyperfine spin states with a Rabi term
coupling. This leads to domain wall solutions having a monotone structure for a
non-cooperative system. We use the moving plane method to prove mono-tonicity
and one-dimensionality of the phase transition solutions. This relies on
totally new estimates for a type of system for which no Maximum Principle a
priori holds. We also derive that one dimensional solutions are unique up to
translations. When the Rabi coefficient is large, we prove that no non-constant
solutions can exist.
