  An integrable field theory, due to path-independence on the space-time plane,
should yield together with an infinite set of independent conserved charges
also similar dual charges determining the boundary and defect contributions. On
the example of the nonlinear Schroedinger equation we unravel hidden
hierarchies and dual structures and show the complete integrability through a
novel Yang-Baxter equation at the classical and quantum level with exact
solution.
