  The direct detection of gravitational waves opens new perspectives for
measuring properties of gravitationally bound compact objects. It is then
important to investigate black holes and neutron stars in alternative theories
of gravity, since they can have features that make them observationally
distinguishable from their General Relativity (GR) counterparts. In this work,
we examine a special case of vector Galileons, a vector-tensor theory of
gravity with interesting cosmological properties, which consists of a one
parameter modification of the Einstein-Maxwell action. Within this theory, we
study configurations describing asymptotically flat, spherically symmetric
black holes and neutron stars. The set of black hole solutions in this theory
is surprisingly rich, generalising results found in GR or in related
scalar-tensor theories. We investigate the properties and conserved charges of
black holes, using both analytical and numerical techniques, highlighting
configurations that are more compact than in GR. We then study properties of
neutron stars, showing how the vector profile can influence the star internal
structure. Depending on properties of matter and fields inside the star,
neutron stars can be more massive than in GR, and they can be even more compact
than Schwarzschild black holes, making these objects observationally
interesting. We also comment on possible extensions of our configurations to
magnetically charged or rotating configurations.
