The magnetic 2-body problem on the sphere

The paper derives the identical reduced equations (5.1) and shows that relative equilibria on the reduced space satisfy m1 = p = 0, leaving two equations in (m2, m3) . It then presents two families of solutions: Type I (existing for all q ≠ π/2, unique up to particle exchange) and Type II (explicit two-branch family with the threshold B = 2√(csc q/(1−cos q)), whose minimum is 4/3^{3/4} at q = 2π/3) . This summary is formalized in Theorem 5.3, parts (1) and (2) . The candidate solution independently reproduces these results by eliminating m2 via ṗ=0 and factoring the remaining equation in m3 (via t = tan(q/2)), identifying one factor with the explicit Type II family (including the same threshold and its minimum) and the other with Type I, for which the two algebraic roots are exchanged by the particle-swap symmetry and no solution exists at q = π/2—precisely matching the paper’s classification and conclusions . The model’s proof route (a clean factorization and a short calculus check of the threshold minimum) differs from the paper’s Mathematica-assisted derivation and symmetry argument (Lemma 5.2 and the involution (5.6)), but the statements and parameter regimes agree.