  We show that one can define a spectral curve for the Cauchy-Riemann operator
on a punctured elliptic curve if one imposes appropriate boundary conditions.
Algebraic curves of the type thus obtained appear as irreducible components of
spectral curves of minimal tori with planar ends in R^3. It appears that these
curves coincide with the spectral curves of certain elliptic KP solitons as
studied by Krichever.
