We consider some examples of canonical transformations of the extended phase space, which map a completely integrable system into another completely integrable system. The proposed transformations are closed to the known Maupertuis-Jacobi mappings and to the Kepler change of the time. Using the Kepler transformation we construct new integrable systems related to the Toda lattices and the Stäckel family of integrable systems.

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Original Russian language edition, entitled "BOKPYf 6EPHCAIIJAA", published by MOCKBA «HAYKA».

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Applying the AdS/CFT correspondence, the expansion of the heavy-quark potential of N = 4 supersymmetric Yang-Mills theory at large N c is carried out to the sub-leading term in the large 't Hooft coupling at zero temperature. The strong coupling corresponds to the semi-classical expansion of the string-sigma model, the gravity dual of the Wilson loop operator, with the sub-leading term expressed in terms of functional determinants of fluctuations. The singularities of these determinants are examined and their contributions are evaluated numerically.