  We study the entanglement spectrum (ES) of a finite XXZ spin 1/2 chain in the
limit \Delta -> -1^+ for both open and periodic boundary conditions. At
\Delta=-1 (ferromagnetic point) the model is equivalent to the Heisenberg
ferromagnet and its degenerate ground state manifold is the SU(2) multiplet
with maximal total spin. Any state in this so-called "symmetric sector" is an
equal weight superposition of all possible spin configurations. In the gapless
phase at \Delta>-1 this property is progressively lost as one moves away from
the \Delta=-1 point. We investigate how the ES obtained from the states in this
manifold reflects this change, using exact diagonalization and Bethe ansatz
calculations. We find that in the limit \Delta ->-1^+ most of the ES levels
show divergent behavior. Moreover, while at \Delta=-1 the ES contains no
information about the boundaries, for \Delta>-1 it depends dramatically on the
choice of boundary conditions. For both open and periodic boundary conditions
the ES exhibits an elegant multiplicity structure for which we conjecture a
combinatorial formula. We also study the entanglement eigenfunctions, i.e. the
eigenfunctions of the reduced density matrix. We find that the eigenfunctions
corresponding to the non diverging levels mimic the behavior of the state
wavefunction, whereas the others show intriguing polynomial structures. Finally
we analyze the distribution of the ES levels as the system is detuned away from
\Delta=-1.
