The limit-point/limit-circle classification for ordinary differential equations with distributional coefficients

We investigate the limit-point/limit-circle classification for the differential equation $J{u}^{\mathrm{\prime }}+qu=\lambda wu$ where $J=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. We identify the situations when the classical alternative of Weyl works and when it fails.