  Highly spinning classical strings on RxS^3 are described by the
Landau-Lifshitz model or equivalently by the Heisenberg ferromagnet in the
thermodynamic limit. The spectrum of this model can be given in terms of
spectral curves. However, it is a priori not clear whether any given admissible
spectral curve can actually be realised as a solution to the discrete Bethe
equations, a property which can be referred to as stability. In order to study
the issue of stability, we find and explore the general two-cut solution or
elliptic curve. It turns out that the moduli space of this elliptic curve shows
a surprisingly rich structure. We present the various cases with illustrations
and thus gain some insight into the features of multi-cut solutions. It appears
that all admissible spectral curves are indeed stable if the branch cuts are
positioned in a suitable, non-trivial fashion.
