Quantum Integrable Systems arising from Separation of Variables on S3

The paper explicitly derives the prolate separated ODEs (Heun type for s1,s3 and a trivial m-equation after s2=cos^2φ), fixes m∈Z, and obtains energy quantization E=D(D+2) via the Heun polynomial truncation condition (α=−d up to symmetry), see Lemma 12 and Lemma 14–15 with the three-term recurrence for Heun polynomials . It identifies a unique focus–focus critical value at (mℏ,λℏ^2)=(0,1) (magenta dot in Fig. 5) and demonstrates quantum monodromy by transporting a unit cell, yielding (v1',v2')=(v1,v2+2v1) when symmetry classes are combined; per single symmetry class the jump is v2→v2+v1, as observed in prior work . The candidate solution reaches the same conclusions but via an ODE-centered argument: Frobenius exponents show only m=0 is resonant at s=1 (hence the defect lies on μ=0), and, for m=0, an exact λ-involution λ↦2E+2−λ is exhibited with unique fixed point λ=E+1, i.e. Λ=1 after scaling. This nicely explains the location (μ,Λ)=(0,1) and the monodromy v2↦v2+v1 per symmetry class, doubling to v2↦v2+2v1 when two symmetry classes are combined. Minor caveats in the model’s writeup (self-adjointness and limit-circle assumptions sketched but not fully justified; mapping the left/right matching across s=1 to the λ-involution is asserted rather than fully proved) do not affect the main claims. Net: paper and model agree on the results; the model supplies a complementary proof mechanism.