  We study Euclidean Wilson loops at strong coupling using the AdS/CFT
correspondence, where the problem is mapped to finding the area of minimal
surfaces in Hyperbolic space. We use a formalism introduced recently by
Kruczenski to perturbatively compute the area corresponding to boundary
contours which are deformations of the circle. Our perturbative expansion is
carried to high orders compared with the wavy approximation and yields new
analytic results. The regularized area is invariant under a one parameter
family of continuous deformations of the boundary contour which are not related
to the global symmetry of the problem. We show that this symmetry of the Wilson
loops breaks at weak coupling at an a priori unexpected order in the
perturbative expansion. We also study the corresponding Lax operator and
algebraic curve for these solutions.
