First boundary-value problem for a class of elliptic systems in a half-space

Using the Fourier transform, we examine the first boundary-value problem for two elliptic systems in a half-space. We prove that for both systems, the homogeneous problem has infinitely many solutions depending on one arbitrary function. At the same time, one of the systems is strongly connected under certain conditions for the coefficients of the system, whereas the second system is always strongly connected.