  We examine the quantum discord between two spins in the exact ground state of
finite spin 1/2 arrays with anisotropic XY couplings in a transverse field B.
It is shown that in the vicinity of the factorizing field B_s, the discord
approaches a common finite non-negligible limit which is independent of the
pair separation and the coupling range. An analytic expression of this limit is
provided. The discord of a mixture of aligned pairs in two different
directions, crucial for the previous results, is analyzed in detail, including
the evaluation of coherence effects, relevant in small samples and responsible
for a parity splitting at B_s. Exact results for finite chains with first
neighbor and full range couplings and their interpretation in terms of such
mixtures are provided.
