  We provide an elementary approach to integrable systems associated with
hyperelliptic curves of infinite genus. In particular, we explore the extent to
which the classical Burchnall-Chaundy theory generalizes in the infinite genus
limit, and systematically study the effect of Darboux transformations for the
KdV hierarchy on such infinite genus curves. Our approach applies to
complex-valued periodic solutions of the KdV hierarchy and naturally identifies
the Riemann surface familiar from standard Floquet theoretic considerations
with a limit of Burchnall-Chaundy curves.
