  There have been recent advances in the construction of algebraic curves for
certain classes of string solutions in the context of the AdS/CFT
correspondence. In this paper we obtain the Lax operators and associated
spectral curves for circular and folded string solutions in
$\,\textrm{AdS}_5\times S^5$. In addition, we provide an original approach for
the reconstruction of string solutions in $\,S^3$ from their corresponding
curves.
