  We give a manifestly invariant definition of the Lagrangian complex germ with
the minimal degree of accuracy required to define the canonical operator. The
equivalence with the traditional definition is proved, and the canonical
operator is constructed in new terms. A new form of the quantization condition
is given, in which the volume form is assumed to be defined on the universal
covering of the Lagrangian manifold rather than on the manifold itself. This
allows one to solve a wider class of eigenvalue problems.
