Linear conjugation problem for elliptic systems in the plane

In an open set $D=\mathbb{C}\setminus\Gamma$ bounded by a Lyapunov contour $\Gamma$ of class $C^{1,\nu}$, we consider the linear conjugation problem for first-order elliptic systems with constant complex and real leading coefficients. Using the integral representation of solutions by a generalized Cauchy-type integral and a generalized Pompeiu integral obtained in this paper, we reduce the original systems to equivalent systems of integral equations. Under certain conditions on the coefficients, the right-hand sides of the systems, and the right-hand side of the boundary condition, using the integral representation obtained and the results of the classical theory of singular operators, we establish a criterion for the Fredholm solvability of the problems posed obtain a formula for the index.