On the integrability of hyperbolic systems of Riccati-type equations

We consider equations of the form U-chi y = U * U-chi, where U(chi, 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-chi and L-y. We define the notion of a Z-graded Lie algebra B corresponding to a given equation. We prove that for every equation under consideration, the corresponding algebra B can be taken as a direct sum of the vector spaces L-chi and L-y if we define the commutators of the elements from L-chi 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-chi and L-y. All of these equations are Darboux-integrable.