On the integrability of hyperbolic systems of riccati-type equations

We consider equations of the form U_xy = U * U_x, 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, L_x and L_y. 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 L_x and L_y if we define the commutators of the elements from L_x and L_y 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 L_x and L_y. All of these equations are Darboux-integrable. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 2, pp. 261–275, November, 1997.     

Symmetries of Systems of the Hyperbolic Riccati Type

Let be a Kac–Moody algebra, U(x,y) be a function defined in , and a be a constant element of . We prove that the equation U _xy = [[U,a],U _x] has two symmetry hierarchies connected by a gauge transformation. In particular, the well-known Konno equation appears in the case of the algebra . The corresponding symmetry hierarchies contain the nonlinear Schrödinger and the Heisenberg magnet equations.