  The symmetries of paths in a manifold $M$ are classified with respect to a
given pointwise proper action of a Lie group $G$ on $M$. Here, paths are
embeddings of a compact interval into $M$. There are at least two types of
symmetries: Firstly, paths that are parts of an integral curve of a fundamental
vector field on $M$ (continuous symmetry). Secondly, paths that can be
decomposed into finitely many pieces, each of which is the translate of some
free segment, where possibly the translate is cut at the two ends of the paths
(discrete symmetry). Here, a free segment is a path $e$ whose $G$-translates
either equal $e$ or intersect it in at most finitely many points. Note that all
the statements above are understood up to the parametrization of the paths. We
will show, for the category of analytic manifolds, that each path is of exactly
one of either types.
  For the proof, we use that the overlap of a path $\gamma$ with one of its
translates is encoded uniquely in a mapping between subsets of $\dom\gamma$.
Running over all translates, these mappings form the so-called
reparametrization set to $\gamma$. It will turn out that, up to conjugation
with a diffeomorphism, any such set is given by the action of a Lie subgroup of
$O(2)$ on $S^1$, restricted in domain and range to some compact interval on
$S^1$. Now, the infinite subgroups correspond to the continuous symmetry above,
finite ones to the discrete symmetry.
