Equality of the Generated First-Integral Field with the Full Annihilator

Determine whether the fields generated by the explicit first integrals \(F_{ij}\) or \(G_{ij}\) in the separated-variable vector-field construction coincide in general with the full field \(W\) of rational first integrals annihilated by the foliation.

Background

For algebraically integrable separated-variable vector fields, the paper constructs explicit families of first integrals: differences of rational antiderivatives when the logarithmic parts vanish, and exponentials of differences of logarithmic antiderivatives in the purely logarithmic case. These functions generate a subfield of the annihilator WW, and the paper establishes that this subfield has the required transcendence degree.

The paper does not establish that the field generated by these displayed first integrals is exactly the entire annihilator; it only states that WW is a separable closure of the indicated generated field. The remark isolates the unresolved equality between the explicitly generated field and WW.

References

I am not sure if the equalities $\overline{\mathbb{Q}(G_{ij})=W$ or $\overline{\mathbb{Q}(F_{ij})=W$ hold in general.

— On Algebraic Integrability of Vector Fields with Rational Function Coefficients that Separate Variables  (2609.00974 - Grabowski, 1 Sep 2026) in Section 4, Remark immediately following Theorem \ref{thm: main theorem n>=2}