  We first deduce the analytical continuation in the complex planes of the
initial and final three-momenta of the Lippmann-Schwinger equation in coupled
or uncoupled partial-wave amplitudes. This result allows us to deduce a master
equation whose solution is the exact discontinuity of the on-shell partial-wave
amplitudes along the left-hand cut. This equation is always a linear
non-singular integral equation whose solution is fixed exclusively by the
knowledge of the potential, applicable to either regular or singular
potentials. The capability of calculating exactly this discontinuity allows one
to settle the exact $N/D $ method in two-body non-relativistic scattering for
coupled and uncoupled waves. We exemplify this new advance in scattering theory
by explicitly checking the agreement between the Lippmann-Schwinger equation
with the corresponding solutions of the exact $N/D$ method for some examples
that involve regular and singular potentials, either attractive or repulsive.
