  In this work we study the microwave photoconductivity of a two-dimensional
electron system (2DES) in the presence of a magnetic field and a
two-dimensional modulation (2D). The model includes the microwave and Landau
contributions in a non-perturbative exact way, the periodic potential is
treated perturbatively. The Landau-Floquet states provide a convenient base
with respect to which the lattice potential becomes time-dependent, inducing
transitions between the Landau-Floquet levels. Based on this formalism, we
provide a Kubo-like formula that takes into account the oscillatory Floquet
structure of the problem. The total longitudinal conductivity and resistivity
exhibit strong oscillations, determined by $\epsilon = \omega / \omega_c$ with
$\omega$ the radiation frequency and $\omega_c$ the cyclotron frequency. The
oscillations follow a pattern with minima centered at $\omega/\omega_c =j +
{1/2} (l-1) + \delta $, and maxima centered at $\omega/\omega_c =j + {1/2}
(l-1) - \delta $, where $j=1,2,3.......$, $\delta \sim 1/5$ is a constant shift
and $l$ is the dominant multipole contribution. Negative resistance states
(NRS) develop as the electron mobility and the intensity of the microwave power
are increased. These NRS appear in a narrow window region of values of the
lattice parameter ($a$), around $a \sim l_B$, where $l_B$ is the magnetic
length. It is proposed that these phenomena may be observed in artificially
fabricated arrays of periodic scatterers at the interface of ultraclean
$GaAs/Al_xGa_{1-x} As$ heterostructures.
