  We propose a field theoretical model that exhibits spontaneous breaking of
the rotational symmetry. The model has a two-dimensional sphere as extra
dimensions of the space-time and consists of a complex scalar field and a
background gauge field. The Dirac monopole, which is invariant under the
rotations of the sphere, is taken as the background field. We show that when
the radius of the sphere is larger than a certain critical radius, the vacuum
expectation value of the scalar field develops vortices, which pin down the
rotational symmetry to lower symmetries. We evaluate the critical radius and
calculate configurations of the vortices by the lowest approximation. The
original model has a $U(1) \times SU(2)$ symmetry and it is broken to U(1),
U(1), D_3 for each case of the monopole number q = 1/2, 1, 3/2, respectively,
where D_3 is the symmetry group of a regular triangle. Moreover, we show that
the vortex configurations are stable against higher corrections of the
perturbative approximation.
