  An answer to the question: Can, in general, the adoption of a given symmetry
induce a further symmetry, which might be hidden at a first level? has been
attempted in the context of differential geometry of locally homogeneous
spaces. Based on E. Cartan's theory of moving frames, a methodology for finding
all symmetries for any n dimensional locally homogeneous space is provided. The
analysis is applied to 3 dimensional spaces, whereby the embedding of them into
a 4 dimensional Lorentzian manifold is examined and special solutions to
Einstein's field equations are recovered. The analysis is mainly of local
character, since the interest is focused on local structures based on
differential equations (and their symmetries), rather than on the implications
of, e.g., the analytic continuation of their solution(s) and their dynamics in
the large.
