Consistent refinement of Bethe strings for spin and electron models and a new non-Bethe solution

Well known Bethe strings in spin chains, delta-electron gas, Hubbard and t-J models are shown to be imprecise, while their consisten refinement along with a new non-Bethe r-string are discovered. Connection with the earlier results is established and the string hypothesis problem is discussed in the light of the present findings.