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We study energy minimization of a continuum Landau-de Gennes energyfunctional for nematic liquid crystals, in three-dimensional axisymmetricdomains and in a restricted class of $\mathbb{S}^1$-equivariant (i.e., axiallysymmetric) configurations. We assume smooth and nonvanishing$\mathbb{S}^1$-equivariant (e.g. homeotropic) Dirichlet boundary conditions anda physically relevant norm constraint (Lyuksyutov constraint) in the interior.Relying on results in \cite{DMP1} in the nonsymmetric setting, we prove partialregularity of minimizers away from a possible finite set of interiorsingularities lying on the symmetry axis. For a suitable class of domains andboundary data we show that for smooth minimizers (torus solutions) the levelsets of the signed biaxiality are generically finite union of tori ofrevolution. Concerning nonsmooth minimizers (split solutions), we characterizetheir asymptotic behavior around any singular point in terms of explicit$\mathbb{S}^1$-equivariant harmonic maps into $\mathbb{S}^4$, whence thegeneric level sets of the signed biaxiality contains invariant topologicalspheres. Finally, in the model case of a nematic droplet, we provide existenceof torus solutions, at least when the boundary data are suitable uniaxialdeformations of the radial anchoring, and existence of split solutions forboundary data which are suitable linearly full harmonic spheres.

Torus-like solutions for the Landau-de Gennes model. Part II: Topology of $\mathbb{S}^1$-equivariant minimizers