  Adiabatic $U(2)$ geometric phases are studied for arbitrary quantum systems
with a three-dimensional Hilbert space. Necessary and sufficient conditions for
the occurrence of the non-Abelian geometrical phases are obtained without
actually solving the full eigenvalue problem for the instantaneous Hamiltonian.
The parameter space of such systems which has the structure of $\xC P^2$ is
explicitly constructed. The results of this article are applicable for
arbitrary multipole interaction Hamiltonians $H=Q^{i_1,\cdots i_n}J_{i_1}\cdots
J_{i_n}$ and their linear combinations for spin $j=1$ systems. In particular it
is shown that the nuclear quadrupole Hamiltonian $H=Q^{ij}J_iJ_j$ does actually
lead to non-Abelian geometric phases for $j=1$. This system, being bosonic, is
time-reversal-invariant. Therefore it cannot support Abelian adiabatic
geometrical phases.
