What are the problems that are in co-NP but not in NP?
i.e, those problems where incorrect strings can be deterministically verified in polynomial time but the correct strings can't be.
co-NP is the set of complements of problems that are in NP. So co-NP contains problems such as non-3-colourability, Boolean unsatisfiability and so on. Most complexity theorists believe that NP$\,\neq\,$co-NP and one consequence of this is that the complement of any NP-complete problem would be in co-NP but not in NP.
Wikipedia has more information on co-NP.
Your question assumes $NP \neq coNP$, which is currently unknown (but generally assumed to be true). If $NP = coNP$, there is no problem satisfying your requirements.
If $NP \neq coNP$, the complement of any NP-complete problem works. You can find a list on Wikipedia.