  We study the behaviour of global minimizers of a continuum Landau-de Gennes
energy functional for nematic liquid crystals, in three-dimensional axially
symmetric domains domains diffeomorphic to a ball (a nematic droplet) and in a
restricted class of $\bbS^1$-equivariant configurations. It is known from our
previous paper \cite{DMP2} that, assuming smooth and uniaxial (e.g.
homeotropic) boundary conditions and a physically relevant norm constraint in
the interior (Lyuksyutov constraint), minimizing configurations are either of
\emph{torus} or of \emph{split} type. Here, starting from a nematic droplet
with the homeotropic boundary condition, we show how singular (split) solutions
or smooth (torus) solutions (or even both) for the Euler-Lagrange equations do
appear as energy minimizers by suitably deforming either the domain or the
boundary data. As a consequence, when minimizers among $\bbS^1$-equivariant
configurations are singular we derive symmetry breaking result for the
minimization among all competitors.
