  Additive symmetric L\'evy noise can induce directed transport of overdamped
particles in a static asymmetric potential. We study, numerically and
analytically, the effect of an additional dichotomous random flashing in such
L\'evy ratchet system. For this purpose we analyze and solve the corresponding
fractional Fokker-Planck equations and we check the results with Langevin
simulations. We study the behavior of the current as function of the stability
index of the L\'evy noise, the noise intensity and the flashing parameters. We
find that flashing allows both to enhance and diminish in a broad range the
static L\'evy ratchet current, depending on the frequencies and asymmetry of
the multiplicative dichotomous noise, and on the additive L\'evy noise
parameters. Our results thus extend those for dichotomous flashing ratchets
with Gaussian noise to the case of broadly distributed noises.
