  The properties of a noncanonical 3D Wigner quantum oscillator, whose position
and momentum operators generate the Lie superalgebra sl(1|3), are further
investigated. Within each state space W(p), p=1,2,..., the energy E_q,
q=0,1,2,3, takes no more than 4 different values. If the oscillator is in a
stationary state \psi_q \in W(p) then measurements of the non-commuting
Cartesian coordinates of the particle are such that their allowed values are
consistent with it being found at a finite number of sites, called ``nests''.
These lie on a sphere centered on the origin of fixed, finite radius \varrho_q.
The nests themselves are at the vertices of a rectangular parallelepiped. In
the typical cases (p>2) the number of nests is 8 for q=0 and 3, and varies from
8 to 24, depending on the state, for q=1 and 2. The number of nests is less in
the atypical cases (p=1,2), but it is never less than two. In certain states in
W(2) (resp. in W(1)) the oscillator is ``polarized'' so that all the nests lie
on a plane (resp. on a line). The particle cannot be localized in any one of
the available nests alone since the coordinates do not commute. The
probabilities of measuring particular values of the coordinates are discussed.
The mean trajectories and the standard deviations of the coordinates and
momenta are computed, and conclusions are drawn about uncertainty relations.
The rotational invariance of the system is also discussed.
