  We briefly review the status quo of the application of integrable systems
techniques to the AdS/CFT correspondence in the large charge approximation, a
rapidly evolving topic. Intricate string and gauge computations of,
respectively, energies and scaling dimensions agree at the one and two-loop
level, but disagree starting from three loops. To add to this pattern, we
present further computations which demonstrate that for folded and circular
spinning strings the full tower of infinitely many hidden commuting charges,
responsible for the integrability, also agrees up to two, but not three, loops.
