  Random one plus two-body hamiltonians invariant with respect to $O({\cal
N}_1) \oplus O({\cal N}_2)$ symmetry in the group-subgroup chains $U({\cal N})
\supset U({\cal N}_1) \oplus U({\cal N}_2) \supset O({\cal N}_1) \oplus O({\cal
N}_2)$ and $U({\cal N}) \supset O({\cal N}) \supset O({\cal N}_1) \oplus
O({\cal N}_2)$ chains of a variety of interacting boson models are used to
investigate the probability of occurrence of a given $(\omega_1 \omega_2)$
irreducible representation (irrep) to be the ground state in even-even nuclei;
$[\omega_1]$ and $[\omega_2]$ are symmetric irreps of $O({\cal N}_1)$ and
$O({\cal N}_2)$ respectively. Numerical results obtained for ${\cal N}_1 \geq
3, {\cal N}_2=1$ and ${\cal N}_1, {\cal N}_2 \geq 3$ situations are well
explained by an extended Hartree-Bose mean-field analysis.
