  We consider a variational two-dimensional Landau-de Gennes model in the
theory of nematic liquid crystals in a disk of radius $R$. We prove that under
a symmetric boundary condition carrying a topological defect of degree
$\frac{k}{2}$ for some given {\bf even} non-zero integer $k$, there are exactly
two minimizers for all large enough $R$. We show that the minimizers do not
inherit the full symmetry structure of the energy functional and the boundary
data. We further show that there are at least five symmetric critical points.
