  We describe the entanglement of two indistinguishable delocalized
spin-$\frac{1}{2}$ particles in the simplest spatial configuration of three
spatial modes with the constraint that at most one particle occupy each mode.
It is show that this is the only number of modes for which maximally entangled
states exist in such a system. The set of entangled states, including the set
of maximally entangled states, is described and different types of entanglement
in terms of Bell-nonlocal correlations for different partitions of the system
are identified. In particular we focus on the entangled states that are
Bell-local for a tri-partition of the system and cannot be described as a
superposition of bi-partite entangled pairs of localized particles. Two
entanglement invariants are constructed and it is shown that all entanglement
monotones are functions of these. Furthermore, the system has a generic
non-trivial local unitary symmetry with a corresponding $2\pi/3$ fractional
topological phase. In addition to this a necessary and sufficient condition for
the existence of maximally entangled states in systems of arbitrary numbers of
delocalized particles with arbitrary spin, where at most one particle can
occupy each mode, is derived.
