  Bose-Einstein condensation of light in 2D is characterized by two classical
fields corresponding to two polarizations of light as well as by the
distribution of dye molecules inducing light thermalization through dipolar
transition. In the case when this transition is triple-degenerate the resulting
field theory for the condensate of light is O(4) symmetric, which precludes
algebraic long range order in 2D at any finite temperature $T$. If the dipolar
degeneracy is removed, then, equilibrium phases with lower symmetries --
O(2)$\times$Z$_2$ and O(2) can emerge. Accordingly, algebraic off diagonal
order of light condensate becomes possible. An orientationsl disorder
introduced by local dipolar anisotropy can destroy algebraic order in
one-photon density matrix while preserving it in the two-photon one. This
represents formation of the condensate of photon pairs.
