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We investigate the monodromy of the Lax connection for classical IIBsuperstrings on AdS_5xS^5. For any solution of the equations of motion wederive a spectral curve of degree 4+4. The curve consists purely of conservedquantities, all gauge degrees of freedom have been eliminated in this form. Themost relevant quantities of the solution, such as its energy, can be expressedthrough certain holomorphic integrals on the curve. This allows for aclassification of finite gap solutions analogous to the general solution ofstrings in flat space. The role of fermions in the context of the algebraiccurve is clarified. Finally, we derive a set of integral equations whichreformulates the algebraic curve as a Riemann-Hilbert problem. They agree withthe planar, one-loop N=4 supersymmetric gauge theory proving the completeagreement of spectra in this approximation.Comment: 56 pages, v2: first local charge derived, minor improvements,  references added, v3: minor corrections, reference added, to appear in  Commun. Math. Phy

The Algebraic Curve of Classical Superstrings on AdS 5×S 5