We study the spectral theory for the first-order system Ju′+qu=wf of differential equations on the real interval (a,b) when J is a constant, invertible skew-Hermitian matrix and q and w are matrices whose entries are distributions of order zero with q Hermitian and w non-negative. Also, we do not pose the definiteness condition often required for the coefficients of the equation. Specifically, we construct minimal and maximal relations, and study self-adjoint restrictions of the maximal relation. For these we determine Green's function and prove the existence of a spectral (or generalized Fourier) transformation. We have a closer look at the special cases when the endpoints of the interval (a,b) are regular as well as the case of a 2×2 system. Two appendices provide necessary details on distributions of order zero and the abstract spectral theory for relations.