  We consider the question which potentials in the action of a (1+1)
dimensional scalar field theory allowing for spontaneous symmetry breaking have
quantum fluctuations corresponding to reflectionless scattering data. The
general problem of restoration from known scattering data is formulated and a
number of explicit examples is given. Only certain sets of reflectionless
scattering data correspond to symmetry breaking and all restored potentials are
similar either to the Phi**4-model or to the sine-Gordon model.
