  We investigate solutions of the elliptic sinh-Gordon equation of spectral
genus g<3. These solutions are parametrized by complex matrix-valued
polynomials called potentials. On the space of these potentials there act two
commuting flows. The orbits of these flows are called Polynomial Killing fields
and are double periodic. The eigenvalues of these matrix-valued polynomials are
preserved along the flows and determine the lattice of periods. We investigate
the level sets of these eigenvalues, which are called isospectral sets, and the
dependence of the period lattice on the isospectral sets. The limiting cases of
spectral genus one and zero are included. Moreover, these limiting cases are
used to construct on every elliptic curve three conformal maps to R^4 which are
constrained Willmore. Finally, the Willmore functional is calculated in
dependence of the conformal class.
