We show how functional relations, which can be considered as a definition of a quantum integrable theory, entail an integral equation that can be extended upon introducing dynamical variables to a Marchenko-like equation. Then, we naturally derive from the latter a classical Lax pair problem. We exemplify our method by focusing on the massive/massless version of the ODE/IM (Ordinary Differential Equations/Integrable Models) correspondence involving the sinh-Gordon/Lioville model, first emerged in the gauge theories and scattering amplitudes/Wilson loops AdS3 context with many moduli/masses, but in a way which reveals its generality. In fact, we give some hints, in the end, to its application to spin chains.