  We provide topological obstructions to the existence of orbit cylinders of
symmetric orbits, for mechanical systems preserved by antisymplectic
involutions (e.g. the restricted three-body problem). Such cylinders induce
continuous paths which do not cross the bifurcation locus of suitable GIT
quotients of the symplectic group, which are branched manifolds whose topology
provide the desired obstructions. Namely, the complement of the corresponding
loci consist of several connected components which we enumerate and explicitly
describe; by construction these cannot be joined by a path induced by an orbit
cylinder. Our construction extends the notions from Krein theory (which only
applies for elliptic orbits), to allow also for the case of symmetric orbits
which are hyperbolic. This gives a general theoretical framework for the study
of stability and bifurcations of symmetric orbits, with a view towards
practical and numerical implementations within the context of space mission
design. We shall explore this in upcoming work.
