Reduction of symbolic first integrals of planar vector fields

Combot’s paper states and proves three reduction algorithms—ReduceRiccati, ReduceLiouvillian, and ReduceDarbouxian—with precise completeness guarantees (except for the explicitly marked Darbouxian exceptional case k∈{2,3,4,6} with no singularities) and concrete steps tied to integrable connections, residues on rational 1-forms, and degree-bounded searches. These results and their conditions are clearly articulated and justified in the paper’s main theorems and proofs (Theorems 1–3) and procedural descriptions . By contrast, the model’s solution conflates dH with d log H in the Darbouxian→rational step, incorrectly claims “integrality of residues after scaling” implies exactness, omits the essential x-dependence/compatibility (integrable connection) in Riccati→lower, and offers an incomplete Liouvillian→lower criterion that would miss rational reductions proven necessary in the paper; it thus lacks completeness and contains key logical misstatements relative to the paper.