Sometimes Killing vector fields in a given spacetime are described as giving information about a symmetry of that particular spacetime solution.
If I look at the requirement of a Killing vector field (is this a sufficient requirement, or is there more?):
$$\nabla_\mu X_\nu + \nabla_\nu X_\mu = 0 $$
This appears purely local, and because General Relativity is a diffeomorphism invariant theory, this makes me wonder if spacetime solutions will necessarily have an infinite number of "local" Killing vector fields (where "local" means the vector field is only non-zero within a finite region of spacetime, and zero outside of this).
Question: what kinds of symmetries (local, global, continuous, discrete, etc.) can be associated with a Killing vector field?
is it possible for a spacetime solution in GR to have none of these symmetries, and so admit zero Killing vector fields?