Canonical systems in $\mathbb{R}^2$ with periodic potentials and vanishing instability intervals

Canonical systems in $\mathbb{R}^2$ with absolutely continuous real symmetric $\pi$-periodic potentials matrices are considered. A through analysis of the discriminant is given along with the indexing and interlacing of the eigenvalues of the periodic, anti-periodic and Dirichlet-type boundary value problems on $[0,\pi]$. The periodic and anti-periodic eigenvalues are characterized in terms of Dirichlet type eigenvalues. It is shown that all instability intervals vanish if and only if the potential is the product of an absolutely continuous real valued function with the identity matrix.