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This is the second in a series of papers on scattering theory forone-dimensional Schr\"odinger operators with Miura potentials admitting aRiccati representation of the form $q=u'+u^2$ for some $u\in L^2(R)$. Weconsider potentials for which there exist `left' and `right' Riccatirepresentatives with prescribed integrability on half-lines. This classincludes all Faddeev--Marchenko potentials in $L^1(R,(1+|x|)dx)$ generatingpositive Schr\"odinger operators as well as many distributional potentials withDirac delta-functions and Coulomb-like singularities. We completely describethe corresponding set of reflection coefficients $r$ and justify the algorithmreconstructing $q$ from $r$.

Inverse scattering on the line for Schr\"odinger operators with Miura potentials, II. Different Riccati representatives