This is the second in a series of papers on scattering theory for
one-dimensional Schr\"odinger operators with Miura potentials admitting a
Riccati representation of the form $q=u'+u^2$ for some $u\in L^2(R)$. We
consider potentials for which there exist `left' and `right' Riccati
representatives with prescribed integrability on half-lines. This class
includes all Faddeev--Marchenko potentials in $L^1(R,(1+|x|)dx)$ generating
positive Schr\"odinger operators as well as many distributional potentials with
Dirac delta-functions and Coulomb-like singularities. We completely describe
the corresponding set of reflection coefficients $r$ and justify the algorithm
reconstructing $q$ from $r$.