  In this paper, we investigate new integrable extensions of two-center Coulomb
systems. We study the most general $n$-dimensional deformation of the
two-center problem by adding arbitrary functions supporting second order
commuting conserved quantities. The system is superintegrable for $n>4$ and,
for certain choices of the arbitrary functions, reduces to known models
previously discovered. Then, based on this extended system, we introduce an
additional integrable generalisation involving Calogero interactions for $n=3$.
In all examples, including the two-center problem, we explicitly present the
complete list of Liouville integrals in terms of second-order integrals of
motion.
