  Transport properties of a two-dimensional electron gas (2DEG) are studied in
the presence of a perpendicular magnetic field $B$, of a {\it weak}
one-dimensional (1D) periodic potential modulation, and of the spin-orbit
interaction (SOI) described only by the Rashba term. In the absence of the
modulation the SOI mixes the spin-up and spin-down states of neighboring Landau
levels into two new, unequally spaced energy branches. The levels of these
branches broaden into bands in the presence of the modulation and their
bandwidths oscillate with the field $B$. Evaluated at the Fermi energy, the
$n$-th level bandwidth of each series has a minimum or vanishes at different
values of the field $B$. In contrast with the 1D-modulated 2DEG without SOI,
for which only one flat-band condition applies, here there are two flat-band
conditions that can change considerably as a function of the SOI strength
$\alpha$ and accordingly influence the transport coefficients of the 2DEG. The
phase and amplitude of the Weiss and Shubnikov-de Haas (SdH) oscillations
depend on the strength $\alpha$. For small values of $\alpha$ both oscillations
show beating patterns. Those of the former are due to the independently
oscillating bandwidths whereas those of the latter are due to modifications of
the density of states, exhibit an even-odd filling factor transition, and are
nearly independent of the modulation strength. For strong values of $\alpha$
the SdH oscillations are split in two.
