  On the two-dimensional non-linear $\Sigma$-model describing a ferromagnet
with Dzyaloshinskii-Moriya interaction, we build three families of exact static
solutions depending on a single Cartesian variable. One of them describes a
clockwise helix configuration, that characterizes the ground state of the
system. A second one corresponds to a counterclockwise helix, representing an
excited state. These two families of solutions are parameterized by a
continuous parameter that depends on the magnetic field and the
Dzyaloshinskii-Moriya coupling. Finally, the third family exists only for
isolated values of the same parameter, corresponding to a discrete family of
Viviani curves on the target sphere. The degeneracy of the resulting spectrum
suggests that an approximate symmetry may emerge at specific values of the
magnetic field, at which additional solutions could then exist.
