  A Poisson system is a Poisson point process and a group action, together
forming a measure-preserving dynamical system. Ornstein and Weiss proved
Poisson systems over many amenable groups were isomorphic in their 1987 paper.
We consider Poisson systems over non-discrete, non-compact, locally compact
Polish groups, and we prove by construction all Poisson systems over such a
group are finitarily isomorphic, producing examples of isomorphisms for
non-amenable group actions. As a corollary, we prove Poisson systems and
products of Poisson systems are finitarily isomorphic.
  For a Poisson system over a group belonging to a slightly more restrictive
class than above, we further prove it splits into two Poisson systems whose
intensities sum to the intensity of the original, generalizing the same result
for Poisson systems over Euclidean space proved by Holroyd, Lyons, and Soo.
