  We examine the entanglement of cyclic spin 1/2 chains with anisotropic XY Z
Heisenberg couplings of arbitrary range at transverse factorizing magnetic
fields. At these fields the system exhibits a degenerate symmetry-breaking
separable ground state (GS). It is shown, however, that the side limits of the
GS pairwise entanglement at these fields are actually non-zero in finite
chains, corresponding such fields to a GS spin-parity transition. These limits
exhibit universal properties like being independent of the pair separation and
interaction range, and are directly related to the magnetization jump.
Illustrative exact results are shown for chains with I) full range and II)
nearest neighbor couplings. Global entanglement properties at such points are
also discussed.
