  We study the effects of elastic anisotropy on the Landau-de Gennes critical
points for nematic liquid crystals, in a square domain. The elastic anisotropy
is captured by a parameter, $L_2$, and the critical points are described by
three degrees of freedom. We analytically construct a symmetric critical point
for all admissible values of $L_2$, which is necessarily globally stable for
small domains i.e., when the square edge length, $\lambda$, is small enough. We
perform asymptotic analyses and numerical studies to discover at least $5$
classes of these symmetric critical points - the $WORS$, $Ring^{\pm}$,
$Constant$ and $pWORS$ solutions, of which the $WORS$, $Ring^+$ and $Constant$
solutions can be stable. Furthermore, we demonstrate that the novel $Constant$
solution is energetically preferable for large $\lambda$ and large $L_2$, and
prove associated stability results that corroborate the stabilising effects of
$L_2$ for reduced Landau-de Gennes critical points. We complement our analysis
with numerically computed bifurcation diagrams for different values of $L_2$,
which illustrate the interplay of elastic anisotropy and geometry for nematic
solution landscapes, at low temperatures.
