  This is the last paper in a series of three in which we have studied the
Peierls substitution in the case of a weak magnetic field. Here we deal with
two $2d$ Bloch eigenvalues which have a conical crossing. It turns out that in
the presence of an almost constant weak magnetic field, the spectrum near the
crossing develops gaps which remind of the Landau levels of an effective
mass-less magnetic Dirac operator.
