  We analyze ground state (GS) factorization in general arrays of spins $s_i$
with $XXZ$ couplings immersed in nonuniform fields. It is shown that an
exceptionally degenerate set of completely separable symmetry-breaking GS's can
arise for a wide range of field configurations, at a quantum critical point
where all GS magnetization plateaus merge. Such configurations include
alternating fields as well as zero bulk field solutions with edge fields only
and intermediate solutions with zero field at specific sites, valid for
$d$-dimensional arrays. The definite magnetization projected GS's at
factorization can be analytically determined and depend only on the exchange
anisotropies, exhibiting critical entanglement properties. We also show that
some factorization compatible field configurations may result in field-induced
frustration and nontrivial behavior at strong fields.
