Solvable Three-Body Problem and Painlevé Conjectures

For a special choice of the three interparticle coupling constants in the three-body version of a many-body problem in the plane that was recently investigated, the general solution of the equations of motion can be written in closed form (and is remarkably simple). We also discuss another analogous three-body problem and obtain two third-order highly nonlinear autonomous ODEs whose general solutions, we conjecture, are entire. In other words, we conjecture that these ODEs feature (a strong version of) the Painlevé property.