Euclidean minimal tori with planar ends and elliptic solitons

A Euclidean minimal torus with planar ends gives rise to an immersed Willmore torus in the conformal 3--sphere $S^3=\R^3\cup \{\infty\}$. The class of Willmore tori obtained this way is given a spectral theoretic characterization as the class of Willmore tori with reducible spectral curve. A spectral curve of this type is necessarily the double of the spectral curve of an elliptic KP soliton. The simplest possible examples of minimal tori with planar ends are related to 1--gap Lam\'e potentials, the simplest non--trivial algebro geometric KdV potentials. If one allows for translational periods, Riemann's "staircase" minimal surfaces appear as other examples related to 1--gap Lam\'e potentials.