  This article examines the properties of positive solutions to fully nonlinear
systems of integral equations involving Hardy and Wolff potentials. The first
part of the paper establishes an optimal existence result and a Liouville type
theorem for the integral systems. Then, the second part examines the decay
rates of positive bound states at infinity. In particular, a complete
characterization of the asymptotic properties of bounded and decaying solutions
is given by showing that such solutions vanish at infinity with two principle
rates: the slow decay rates and the fast decay rates. In fact, the two rates
can be fully distinguished by an integrability criterion. As an application,
the results are shown to carry over to certain systems of quasilinear
equations.
