We pose a new problem of collinear central configuration in Newtonian $n$-body problem. For a given two-body, we ask whether we can add a new body in a way such that i) the configuration of the total three-body is also collinear central with the configuration of the initial two-body being fixed and further ii) the initial two-body keeps its motion without any change during the process. We find three solutions to the above problem. We also consider a similar problem such that while the condition i) is satisfied but by modifying the condition ii) the motion of the initial two-body is not necessarily equal to the original one. We also find explicit solutions to the second problem.
Naoko Yoshimi. Akira Yoshioka. "2+1-Moulton Configuration." J. Geom. Symmetry Phys. 50 79 - 95, 2018. https://doi.org/10.7546/jgsp-50-2018-79-95