  Transport properties of the two-dimensional electron gas (2DEG) are
considered in the presence of a perpendicular magnetic field $B$ and of a {\it
weak} two-dimensional (2D) periodic potential modulation in the 2DEG plane. The
symmetry of the latter is rectangular or hexagonal. The well-known solution of
the corresponding tight-binding equation shows that each Landau level splits
into several subbands when a rational number of flux quanta $h/e$ pierces the
unit cell and that the corresponding gaps are exponentially small. Assuming the
latter are closed due to disorder gives analytical wave functions and
simplifies considerably the evaluation of the magnetoresistivity tensor
$\rho_{\mu\nu}$. The relative phase of the oscillations in $\rho_{xx}$ and
$\rho_{yy}$ depends on the modulation periods involved. For a 2D modulation
with a {\bf short} period $\leq 100$ nm, in addition to the Weiss oscillations
the collisional contribution to the conductivity and consequently the tensor
$\rho_{\mu\nu}$ show {\it prominent peaks when one flux quantum $h/e$ passes
through an integral number of unit cells} in good agreement with recent
experiments. For periods $300- 400$ nm long used in early experiments, these
peaks occur at fields 10-25 times smaller than those of the Weiss oscillations
and are not resolved.
