  The multiplier spectral curve of a conformal torus in the 4-sphere is
essentially, see arXiv:0712.2311, given by all Darboux transforms of the
conformal torus. In the particular case when the conformal immersion is a
Hamiltonian stationary torus in Euclidean 4-space, the left normal of the
immersion is harmonic, hence we can associate a second Riemann surface: the
eigenline spectral curve of the left normal, as defined by Hitchin. We show
that the multiplier spectral curve of a Hamiltonian stationary torus and the
eigenline spectral curve of its left normal are biholomorphic Riemann surfaces
of genus zero. Moreover, we prove that all Darboux transforms, which arise from
generic points on the spectral curve, are Hamiltonian stationary whereas we
also provide examples of Darboux transforms which are not even Lagrangian.
