  The effect of a modulated magnetic field on the electronic structure of
neutral graphene is examined in this paper. It is found that application of a
small staggered modulated magnetic field does not destroy the Dirac-cone
structure of graphene and so preserves its 4-fold zero-energy degeneracy. The
original Dirac points (DPs) are just shifted to other positions in k space. By
varying the staggered field gradually, new DPs with exactly the same
electron-hole crossing energy as that of the original DPs, are generated, and
both the new and original DPs are moving continuously. Once two DPs are shifted
to the same position, they annihilate each other and vanish. The process of
generation and evolution of these DPs with the staggered field is found to have
a very interesting patten, which is examined carefully. Generally, there exists
a corresponding branch of anisotropic massless fermions for each pair of DPs,
resulting in that each Landau level (LL) is still 4-fold degenerate except the
zeroth LL which has a robust $4n_t$-fold degeneracy with nt the number of pairs
of DPs. As a result, the Hall conductivity $\sigma_{xy}$ shows a step of size
$4n_te^2/h$ across zero energy.
