Let M be the set of real \(3\times 3\) matrices Q with \(Tr(Q)=0\) and \(Tr(Q^ 2)=1\), and let O(3) act on M by conjugation. The projective plane \({\mathbb{R}}P^ 2\) is equivariantly embedded in M by mapping each point \(\pm (x_ 1,x_ 2,x_ 3)\) to the matrix with entries \(q_{ij}=(3x_ ix_ j-\delta_{ij})/\sqrt{6}.\) A uniaxial nematic liquid crystal with underlying space domain X is geometrically described by a function \(X\to {\mathbb{R}}P^ 2\), and a biaxial one by a function \(X\to M\). The author announces several results describing the sets of equivariant homotopy classes of maps \(S^ 2\to {\mathbb{R}}P^ 2\) and \((D^ 3,S^ 2)\to (M,{\mathbb{R}}P^ 2)\), for specific finite subgroups of O(3), thus giving a local symmetric topological classification of singularities in nematic liquid crystals.