  A complete and rigorous determination of the possible ground states for
D-wave pairing Bose condensates is presented, using a geometrical invariant
theory approach to the problem. The order parameter is argued to be a vector,
transforming according to a ten dimensional real representation of the group
$G=${\bf O}$_3\otimes${\bf U}$_1\times <{\cal T}>$. We determine the equalities
and inequalities defining the orbit space of this linear group and its symmetry
strata, which are in a one-to-one correspondence with the possible distinct
phases of the system. We find 15 allowed phases (besides the unbroken one),
with different symmetries, that we thoroughly determine. The group-subgroup
relations between bordering phases are pointed out. The perturbative sixth
degree corrections to the minimum of a fourth degree polynomial $G$-invariant
free energy, calculated by Mermin, are also determined.
