| Summary: | We consider the two- and <i>n</i>-body problems on the two-dimensional conformal sphere <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">M</mi><mi>R</mi><mn>2</mn></msubsup></semantics></math></inline-formula>, with a radius <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. We employ an alternative potential free of singularities at antipodal points. We study the limit of relative equilibria under the SO(2) symmetry; we examine the specific conditions under which a pair of positive-mass particles, situated at antipodal points, can maintain a state of relative equilibrium as they traverse along a geodesic. It is identified that, under an appropriate radius–mass relationship, these particles experience an unrestricted and free movement in alignment with the geodesic of the canonical Killing vector field in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">M</mi><mi>R</mi><mn>2</mn></msubsup></semantics></math></inline-formula>. An even number of bodies with pairwise conjugated positions, arranged in a regular <i>n</i>-gon, all with the same mass <i>m</i>, move freely on a geodesic with suitable velocities, where this geodesic motion behaves like a relative equilibrium. Also, a center of mass formula is included. A relation is found for the relative equilibrium in the two-body problem in the sphere similar to the Snell law.
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