Let \(D\) be a simply connected domain in the complex plane with the boundary \(\partial D=\Gamma \cup [-1,1]\). The quasilinear degenerate elliptic system \(H(y)u_x - v_y =a_1 u+ b_1v +c_1\), \(H(y)v_x+u_y =a_2 u+ b_2v +c_2\) with \(H(0)=0\) and discontinuous coefficients is considered in \(D\). Following [I. N. Vekua, Generalized analytic functions. (International Series of Monographs on Pure and Applied Mathematics. Vol. 25.) Oxford-London-New York-Paris: Pergamon Press; Reading, Mass.- London: Addison-Wesley (1962; Zbl 0100.07603)] the authors deduce the representation \(w=u+iv = \Phi (Z) e^{ \phi (Z) } + \psi (Z) \), where \(Z = x +i \int_0^y H(t) \,dt\), \(\Phi (Z)\) is analytic on \(Z=Z(z)\) (\(z=x+i y \in D\)), \(\phi\) and \(\psi\) are written in the form of double singular integrals. The existence and uniqueness of solutions of the Riemann–Hilbert problem \(\operatorname{Re}[\overline{\lambda(t)} w(t)] = r(t)\), \(t \in \partial D\), is discussed.