Self-adjoint extensions of symmetric relations associated with systems of ordinary differential equations with distributional coefficients

We study the extension theory for the two-dimensional first-order system $Ju' +qu = wf$ of differential equations on the real interval $(a,b)$ where $J$ is a constant, invertible, skew-hermitian matrix and $q$ and $w$ are matrices whose entries are real distributions of order $0$ with $q$ hermitian and $w$ non-negative. Specifically, we characterize the boundary conditions for solutions $u$ in the closure of the minimal relation, as well as describe the properties of quasi-boundary conditions which yield self-adjoint extensions. We then apply these ideas to a popular extension of non-negative minimal relations: the Krein-von Neumann extension. For more context on how the Krein-von Neumann is defined, an appendix shows a construction of the Friedrichs extension from which the Krein-von Neumann is traditionally defined.