Abstract
The ordered states of a planar antiferromagnet with a triangular lattice are investigated in the presence of a magnetic field. The spin wave free energy is taken into account and proved to be important for determining the properties of the system. The phase diagram is constructed. It contains four different phases with rigorous long-range order. Three of them are characterised by different configurations of mean magnetic moments for the three substances. The existence of one or more non-trivial phase with an algebraic decay of the correlation functions is very probable.