Well-Characterized vs NP-Complete Definition: Well-Characterized also known as Good-Characterized is defined as follows:


  Problem that have Yes and No certificates i.e., are in NP $\cap$ co-NP, are
  said to be Well-Characterized. (Reference: V.Vazirani,
  Approximation Algorithm)


Does exist any Well-Chracterized problem that is also being NP-Complete? I don't looking for any proof, I just want a real-world example of such problem.
We don't know. If any NP-complete problem were in co-NP, then we'd have NP$\,=\,$co-NP and we don't know whether that's true or not.