On the spectral theory of systems of first order equations with periodic distributional coefficients

We establish a Floquet theorem for a first-order system of differential equations $u^{\mathrm{\prime }}=ru$ where $r$ is an $n\times n$-matrix whose entries are periodic distributions of order $0$. Then we investigate, when $n=1$ and $n=2$, the spectral theory for the equation $J{u}^{\mathrm{\prime }}+qu=wf$ on $\mathbb{R}$ when $J$ is a real, constant, invertible, skew-symmetric matrix and $q$ and $w$ are periodic matrices whose entries are real distributions of order $0$ with $q$ symmetric and $w$ non-negative.