  We present a theory of Sturm-Liouville non-symmetric vessels, realizing an
inverse scattering theory for the Sturm-Liouville operator with analytic
potentials on the line. This construction is equivalent to the construction of
a matrix spectral measure for the Sturm-Liouville operator, defined with an
analytic potential on the line. Evolving such vessels we generate KdV vessels,
realizing solutions of the KdV equation. As a consequence, we prove the
following theorem:
  Suppose that q(x) is an analytic function on R. There exists a KdV vessel,
which exists on a subset O of the plane. For each real x there exists positive
T_x such that $\{x\}\times [-T_x,T_x]$ is in O. The potential q(x) is realized
by the vessel for t=0.
  Since we also show that if q(x,t) is a solution of the KdV equation on a
strip $R\times[0,T]$, then there exists a vessel, realizing it, the theory of
vessels becomes a universal tool to study this problem. Finally, we notice that
the idea of the proof applies to a similar existence of a solution for
evolutionary NLS and Boussinesq equations, since both of these equations
possess vessel constructions.
