  A normal form transformation is carried out on the operators of a complete
set of commuting observables in a multidimensional, integrable quantum system,
mapping them by unitary conjugation into functions of the harmonic oscillators
in the various degrees of freedom. The transformation works at the level of the
Weyl symbols of the operators, which are manipulated as formal power series in
hbar by use of the Moyal star product. It is assumed that the Weyl symbol of
one of the operators (the Hamiltonian) has a generic, stable fixed point in
phase space. The normal form transformation takes place in a neighborhood of
this fixed point. Once the normal form has been achieved, the
Einstein-Brillouin-Keller or torus quantization rule follows easily, including
higher order corrections in hbar. Crucial parts of the normal form
transformation are not obvious generalizations of the one-dimensional case, nor
is final quantization rule. The result raises some issues of differential
geometry not found in the one-dimensional case.
