  The Symmetric Informationally Complete Positive Operator-Valued Measures
(SIC-POVMs) are known to exist in all dimensions $\leq 151$ and few higher
dimensions as high as $1155$. All known solutions with the exception of the
Hoggar solutions are covariant with respect to the Weyl-Heisenberg group and in
the case of dimension 3 it has been proven that all SIC-POVMs are
Weyl-Heisenberg group covariant. In this work, we introduce two functions with
which SIC-POVM Gram matrices can be generated without the group covariance
constraint. We show analytically that the SIC-POVM Gram matrices exist on
critical points of surfaces formed by the two functions on a subspace of
symmetric matrices and we show numerically that in dimensions 4 to 7, all
SIC-POVM Gram matrices lie in disjoint solution "islands". We generate
$O(10^6)$ and $O(10^5)$ Gram matrices in dimensions 4 and 5, respectively and
$O(10^2)$ Gram matrices in dimensions 6 and 7. For every Gram matrix obtained,
we generate the symmetry groups and show that all symmetry groups contain a
subgroup of $3n^2$ elements. The elements of the subgroup correspond to the
Weyl-Heisenberg group matrices and the order-3 unitaries that generate them.
All constructed Gram matrices have a unique generating set. Using this fact, we
generate permutation matrices to map the Gram matrices to known Weyl-Heisenberg
group covariant solutions. In dimensions 4 and 5, the absence of a solution
with a smaller symmetry, strongly suggests that non-group covariant SIC-POVMs
cannot be constructed.
