  Among the stationary configurations of the Hamiltonian of a classical O$(n)$
lattice spin model, a class can be identified which is in one-to-one
correspondence with all the the configurations of an Ising model defined on the
same lattice and with the same interactions. Starting from this observation it
has been recently proposed that the microcanonical density of states of an
O$(n)$ model could be written in terms of the density of states of the
corresponding Ising model. Later, it has been shown that a relation of this
kind holds exactly for two solvable models, the mean-field and the
one-dimensional $XY$ model, respectively. We apply the same strategy to derive
explicit, albeit approximate, expressions for the density of states of the
two-dimensional $XY$ model with nearest-neighbor interactions on a square
lattice. The caloric curve and the specific heat as a function of the energy
density are calculated and compared against simulation data, yielding a very
good agreement over the entire energy density range. The concepts and methods
involved in the approximations presented here are valid in principle for any
O$(n)$ model.
