The algebraic curve (finite-gap) classification of rotating string solutions was very important in the development of integrability through comparison with analogous structures at weak coupling. The classification was based on the analysis of monodromy around the closed string cylinder. In this paper we show that certain classical Wilson loop minimal surfaces corresponding to the null cusp and qq potential with trivial monodromy can, nevertheless, be described by appropriate algebraic curves. We also show how a correlation function of a circular Wilson loop with a local operator fits into this framework. The latter solution has identical monodromy to the pointlike BMN string and yet is significantly different. *

We identify classical string solutions which directly give the classical part of the strong coupling pomeron intercept. The relevant solution is a close cousin of the GKP folded string, which is not surprising given the known relation with twist-2 operators. Our methods are applicable, however, also for nonzero conformal spin where we do not have a clear link with anomalous dimensions of a concrete class of operators. We analyze the BFKL folded string from the algebraic curve perspective and investigate its possible particle content.