  We consider a slowly decaying oscillatory potential such that the
corresponding 1D Schr\"odinger operator has a positive eigenvalue embedded into
the absolutely continuous spectrum. This potential does not fall into a known
class of initial data for which the Cauchy problem for the Korteweg-de Vries
(KdV) equation can be solved by the inverse scattering transform. We
nevertheless show that the KdV equation with our potential does admit a closed
form classical solution in terms of Hankel operators. Comparing with rapidly
decaying initial data our solution gains a new term responsible for the
positive eigenvalue. To some extend this term resembles a positon (singular)
solution but remains bounded. Our approach is based upon certain limiting
arguments and techniques of Hankel operators.
