Zero-curvature representation for a chiral-type three-field system

The matrix 4 x 4 zero-curvature representation for a two-dimensional chiral-type system with three fields is constructed. The system under consideration belongs to the class of scalar fields with the Lagrangian L = 1/2 g(ij)(u)u(x)(i)u(t)(j) + f(u), where g(ij) is the metric tensor of the three-dimensional reducible Riemann space. This system was found by the authors earlier in the frame of the symmetry method. The zero-curvature representation is computed with the help of the third order symmetry u(t) = S(u). This was possible because the hyperbolic system is a nonlocal member in the hierarchy of the evolution systems and the matrix U of the zero-curvature representation is the common one for the whole hierarchy. As the test for non-triviality of the representation the recursion relations for the conserved currents are found.