For meromorphic KdV potentials, the Schrödinger equation in the Lax pair is solvable by quadrature if and only if the potential is reflectionless, but the paper proves this only under extra hypotheses on the scattering coefficient and on analyticity at infinity.
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Solvability of the Korteweg-de Vries equation under meromorphic initial conditions by quadrature
For meromorphic KdV potentials, the Schrödinger equation in the Lax pair is solvable by quadrature if and only if the potential is reflectionless, but the paper proves this only under extra hypotheses on the scattering coefficient and on analyticity at infinity.