Symmetric periodic solutions in the generalized Sitnikov problem with homotopy methods

The paper proves Theorem 1—existence, for each integer q > 1/√β and each p ≤ ⌊√β q⌋, of 2πq-periodic z-axis motions that are anti-periodic with half-period πq, even in time, and have exactly 2p zeros—via a homotopy/Leray–Schauder degree continuation from the conservative λ=0 case through λ∈[0,1] (defining the Hj(t;λ) homotopy, restricting to the symmetric subspace Y, obtaining a priori bounds, excluding branch collisions except at the trivial solution, and building a neighborhood of the origin where the relevant branches cannot end by Sturm–Liouville analysis) . The candidate solution sketches a different, standard route using the area-preserving Poincaré map over time πq composed with central inversion, twist monotonicity via Prüfer variables, and Poincaré–Birkhoff to produce fixed points with the required symmetry and zero count. While this approach matches the statement and relies on well-known tools, the sketch omits technical verifications (invariance of an annulus for the map, full twist conditions on the boundary, and an a priori bound) that the paper handles rigorously. Thus, both establish the same result, but the paper’s proof is complete and the model’s is a high-level sketch consistent with the literature.