  We briefly review the most relevant aspects of complete integrability for
classical systems and identify those aspects which should be present in a
definition of quantum integrability.
  We show that a naive extension of classical concepts to the quantum framework
would not work because all infinite dimensional Hilbert spaces are unitarily
isomorphic and, as a consequence, it would not be easy to define degrees of
freedom. We argue that a geometrical formulation of quantum mechanics might
provide a way out.
