  The problem of two fixed centers was introduced by Euler as early as in 1760.
It plays an important role both in celestial mechanics and in the microscopic
world. In the present paper we study the spatial problem in the case of
arbitrary (both positive and negative) strengths of the centers. Combining
techniques from scattering theory and Liouville integrability, we show that
this spatial problem has topologically non-trivial scattering dynamics, which
we identify as scattering monodromy. The approach that we introduce in this
paper applies more generally to scattering systems that are integrable in the
Liouville sense.
