  In this work, we perform a first study of basic invariant sets of the spatial
Hill's four-body problem, where we have used both analytical and numerical
approaches. This system depends on a mass parameter mu in such a way that the
classical Hill's problem is recovered when mu = 0. Regarding the numerical
work, we perform a numerical continuation, for the Jacobi constant C and
several values of the mass parameter mu by applying a classical
predictor-corrector method, together with a high-order Taylor method
considering variable step and order and automatic differentiation techniques,
to specific boundary value problems related with the reversing symmetries of
the system. The solution of these boundary value problems defines initial
conditions of symmetric periodic orbits. Some of the results were obtained
departing from periodic orbits within Hill's three-body problem. The numerical
explorations reveal that a second distant disturbing body has a relevant effect
on the stability of the orbits and bifurcations among these families. We have
also found some new families of periodic orbits that do not exist in the
classical Hill's three-body problem; these families have some desirable
properties from a practical point of view.
