Solvability of the Korteweg-de Vries equation under meromorphic initial conditions by quadrature

The paper proves the two directions under (A1): (i) reflectionless plus a zero of a(k) in C+ implies solvability by quadrature for all k (Theorem 1.3), via a finite system obtained from a contour/projection argument around the zeros of a(k) and the reflectionless condition b≡0, yielding Jost solutions as rational combinations of exponentials ; and (ii) if u is analytic at ∞ and the ODE is Liouvillian for all k, then b(k)=0 on R*, by relating Stokes matrices at y=0 (x=∞) to the scattering matrix and invoking the Ramis density theorem (Theorem A.2) together with the SL2(C) subgroup classification (Proposition 2.3) and an identity-theorem argument in k . The model’s argument reaches the same conclusions using a Kay–Moses/GLM-plus-Darboux route for (i) and a Galois–Stokes triangularization argument for (ii). Minor omissions in the model (e.g., not remarking on reflectionless cases with a(k)≡1 beyond u≡0, noted in the paper’s Remark 1.5(i) , and not detailing confluent Darboux for multiple zeros) do not alter correctness. Hence, both are correct but the proofs are substantially different.