Integrable extensions of two-center Coulomb systems

In this paper, we investigate new integrable extensions of two-center Coulomb systems. We study the most general <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>n</a:mi></a:math>-dimensional deformation of the two-center problem by adding arbitrary functions supporting second-order commuting conserved quantities. The system is superintegrable for <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>n</c:mi><c:mo>></c:mo><c:mn>4</c:mn></c:math> 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 generalization involving Calogero interactions for <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>n</e:mi><e:mo>=</e:mo><e:mn>3</e:mn></e:math>. 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.