  We study a class of symmetric critical points in a variational $2D$ Landau -
de Gennes model where the state of nematic liquid crystals is described by
symmetric traceless $3\times 3$ matrices. These critical points play the role
of topological point defects carrying a degree $\frac k 2$ for a nonzero
integer $k$. We prove existence and study the qualitative behavior of these
symmetric solutions. Our main result is the instability of critical points when
$k\neq \pm 1, 0$.
