  We derive the Lax operator for a very large family of classical minimal
surface solutions in $AdS_3$ describing Wilson loops in $\mathcal{N}=4$ SYM
theory. These solutions, constructed by Ishizeki, Kruczenski and Ziama, are
associated with a hyperellictic surface of odd genus. We verify that the
algebraic curve derived from the Lax operator is indeed none-other than this
hyperelliptic surface.
