  Studying the physics of compact objects in modified theories of gravity is
important for understanding how future observations can test alternatives to
General Relativity. We consider a subset of vector-tensor Galileon theories of
gravity characterized by new symmetries, which can prevent the propagation of
the vector longitudinal polarization, even in absence of Abelian gauge
invariance. We investigate new spherically symmetric and slowly rotating
solutions for these systems, including an arbitrary matter Lagrangian. We show
that, under certain conditions, there always exist stealth configurations whose
geometry coincides with solutions of Einstein gravity coupled with the
additional matter. Such solutions have a non-trivial profile for the vector
field, characterized by independent integration constants, which extends to
asymptotic infinity. We interpret our findings in terms of the symmetries and
features of the original vector-tensor action, and on the number of degrees of
freedom that it propagates. These results are important to eventually describe
gravitationally bound configurations in modified theories of gravity, such as
black holes and neutron stars, including realistic matter fields forming or
surrounding the object.
