  A quasiclassical approximation is constructed to describe the eigenvalues of
the magnetic Laplacian on a compact Riemannian manifold in the case when the
magnetic field is not given by an exact 2-form. For this, the multidimensional
WKB method in the form of Maslov canonical operator is applied. In this case,
the canonical operator takes values in sections of a nontrivial line bundle.
The constructed approximation is demonstrated for the Dirac magnetic monopole
on the two-dimensional sphere.
