  Localized magnons states, due to flat bands in the spectrum, is an intensely
studied phenomenon and can be found in many frustrated magnets of different
spatial dimensionality. The presence of Dzyaloshinskii-Moriya (DM) interactions
may change radically the behavior in such systems. In this context, we study a
paradigmatic example of a one-dimensional frustrated antiferromagnet, the
sawtooth chain in the presence of DM interactions. Using both path integrals
methods and numerical Density Matrix Renormalization Group, we revisit the
physics of localized magnons and determine the consequences of the DM
interaction on the ground state. We have studied the spin current behavior,
finding three different regimes. First, a Luttinger-liquid regime where the
spin current shows a step behavior as a function of parameter $D$, at a low
magnetic field. Increasing the magnetic field, the system is in the Meissner
phase at the $m = 1/2$ plateau, where the spin current is proportional to the
DM parameter. Finally, further increasing the magnetic field and for finite $D$
there is a small stiffness regime where the spin current shows, at fixed
magnetization, a jump to large values at $D = 0$, a phenomenon also due to the
flat band.
