  We examine the existence of completely separable ground states (GS) in finite
spin-$s$ arrays with anisotropic $XYZ$ couplings, immersed in a non-uniform
magnetic field along one of the principal axes. The general conditions for
their existence are determined. The analytic expressions for the separability
curve in field space, and for the ensuing factorized state and GS energy, are
then derived for alternating solutions, valid for any spin and size. They
generalize results for uniform fields and show that nonuniform fields can
induce GS factorization also in systems which do not exhibit this phenomenon in
a uniform field. It is also shown that such curve corresponds to a fundamental
$S_z$-parity transition of the GS, present for any spin and size, and that two
different types of GS parity diagrams can emerge, according to the anisotropy.
The role of factorization in the magnetization and entanglement of these
systems is also analyzed, and analytic expressions at the borders of the
factorizing curve are provided. Illustrative examples for spin pairs and chains
are as well discussed.

Quantum Physics;Other Condensed Matter