Abstract
We consider equations of the form Uxy = U * Ux, where U(x, y) is a function taking values in an arbitrary finite-dimensional algebra T over the field ℂ. We show that every such equation can be naturally associated with two characteristic Lie algebras, Lx and Ly. We define the notion of a ℤ-graded Lie algebraB corresponding to a given equation. We prove that for every equation under consideration, the corresponding algebraB can be taken as a direct sum of the vector spaces Lx and Ly if we define the commutators of the elements from Lx and Ly by means of the zero-curvature relations. Assuming that the algebra T has no left ideals, we classify the equations of the specified type associated with the finite-dimensional characteristic Lie algebras Lx and Ly. All of these equations are Darboux-integrable.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 2, pp. 261–275, November, 1997.
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Bormisov, A.A., Gudkova, E.S. & Mukminov, F.K. On the integrability of hyperbolic systems of riccati-type equations. Theor Math Phys 113, 1418–1430 (1997). https://doi.org/10.1007/BF02634167
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DOI: https://doi.org/10.1007/BF02634167