Spectral curves for Cauchy-Riemann operators on punctured elliptic curves

We show that one can define a spectral curve for the Cauchy-Riemann operator on a punctured elliptic curve under appropriate boundary conditions. The algebraic curves thus obtained arise, for example, as irreducible components of the spectral curves of minimal tori with planar ends in ?³. It turns out that these curves coincide with the spectral curves of certain elliptic KP solitons studied by Krichever.