  Every (full) finite Gabor system generated by a unit-norm vector $g\in
\mathbb{C}^d$ is a finite unit-norm tight frame (FUNTF), and can thus be
associated with a (Gabor) positive operator valued measure (POVM). Such a POVM
is informationally complete if the $d^2$ corresponding rank one matrices form a
basis for the space of $d\times d$ matrices. A sufficient condition for this to
happen is that the POVM is symmetric, which is equivalent to the fact that the
associated Gabor frame is an equiangular tight frame (ETF). The existence of
Gabor ETF is an important special case of the Zauner conjecture. It is known
that generically all Gabor FUNTFs lead to informationally complete POVMs. In
this paper, we initiate a classification of non-complete Gabor POVMs. In the
process we establish some seemingly simple facts about the eigenvalues of the
Gram matrix of the rank one matrices generated by a finite Gabor frame. We also
use these results to construct some sets of $d^2$ unit vectors in
$\mathbb{C}^d$ with a relatively smaller number of distinct inner products.
