Vector functional-difference equation in electromagnetic scattering

A vector functional-difference equation of the first order with a special matrix coefficient is analysed. It is shown how it can be converted into a Riemann-Hilbert boundary-value problem on a union of two segments on a hyperelliptic surface. The genus of the surface is defined by the number of zeros and poles of odd order of a characteristic function in a strip. An even solution of a symmetric Riemann-Hilbert problem is also constructed. This is a key step in the procedure for diffraction problems. The proposed technique is applied for solving in closed form a new model problem of electromagnetic scattering of a plane wave obliquely incident on an anisotropic impedance half-plane (all the four impedances are assumed to be arbitrary).