We present a complete list of nonlinear one-field hyperbolic equations with integrable third-order x and y symmetries. The list includes both equations of the sine-Gordon type and equations linearizable by differential substitutions.

Abstract. V.V. Sokolov's modifying symmetry approach is applied to anisotropic evolution equations of the third order on the n-dimensional sphere. The main result is a complete classification of such equations. Auto-Bäcklund transformations are also found for all equations.

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