  We propose in this work a concept of integrability for quantum systems, which
corresponds to the concept of noncommutative integrability for systems in
classical mechanics. We determine a condition for quantum operators which can
be a suitable replacement for the condition of functional independence for
functions on the classical phase space. This condition is based on the
properties of the main parts of the operators with respect to the momenta. We
are led in this way to the definition of what we call a "quasi-integrable
quantum system". This concept will be further developed in a series of
following papers.
