A COMPLEX LIMIT CYCLE NOT INTERSECTING THE REAL PLANE

The paper proves that the polynomial vector field z' = w + z(z^2 + w^2 + 1), w' = −z + w(z^2 + w^2 + 1) admits the algebraic curve L: z^2 + w^2 + 1 = 0 as a complex limit cycle not intersecting R^2, using the Khanehdani–Suwa first-variation formula for holonomy h'_γ(0) = exp ∮_γ α (definitions and theorem stated in the paper’s preliminaries). See the main theorem and set-up (; holonomy definitions and the first-variation formula in ), and the computation showing L is an invariant nonsingular leaf (). However, in restricting α to L and computing its integral along the loop via the parametrization φ, the paper drops a factor of 2 and makes an algebraic sign/simplification error, yielding ∮_γ α = −2π and h'_γ(0) = e^{−2π} (; ). Correcting these slips gives ∮_γ α = ±4π (orientation dependent), so |h'_γ(0)| = e^{4π} and, with flow orientation, h'_γ(0) = e^{−4π}, which matches the model’s direct computation via the scalar transverse ODE F' = 2F(F−1). The model’s solution is correct, clean, and quantitatively consistent; the paper’s conclusion (nontrivial holonomy) is correct despite minor arithmetic errors.