  We analyze exact ground state (GS) separability in general $N$ particle
systems with two-site couplings. General necessary and sufficient conditions
for full separability, in the form of one and two-site eigenvalue equations,
are first derived. The formalism is then applied to a class of $SU(n)$-type
interacting systems, where each constituent has access to $n$ local levels, and
where the total number parity of each level is preserved. Explicit
factorization conditions for parity-breaking GS's are obtained, which
generalize those for $XYZ$ spin systems and correspond to a fundamental GS
multilevel parity transition where the lowest $2^{n-1}$ energy levels cross. We
also identify a multicritical factorization point with exceptional high
degeneracy proportional to $N^{n-1}$, arising when the total occupation number
of each level is preserved, in which any uniform product state is an exact GS.
Critical entanglement properties (like full range pairwise entanglement) are
shown to emerge in the immediate vicinity of factorization. Illustrative
examples are provided.
