Abstract In this paper we investigate the analytical properties of systems of linear ordinary differential equations (ODEs) with unsmooth nonintegrable inhomogeneities and a time singularity of the first kind. We are especially interested in specifying the structure of general linear two-point boundary conditions guaranteeing existence and uniqueness of solutions which are continuous on a closed interval including the singular point. Moreover, we study the convergence behavior of collocation schemes applied to solving the problem numerically. Our theoretical results are supported by numerical experiments. MSC: 34A12, 34A30, 34B05.