A generalisation is given of the inverse problem considered in S. Currie, B.A. Watson, and T.T. Roth. First‐order systems in on with periodic matrix potentials and vanishing instability intervals, Math. Meth. Appl. Sci.38 (2015), 4435‐4447. In particular, the self‐adjoint first‐order system, JY ′ + QY = λY, with integrable, real, symmetric, and π‐periodic, 2 × 2 matrix potential Q is considered, where . It is shown that all eigenvalues to the above equation with boundary conditions Y(π) = ±R(θ)Y(0), where R(θ) is the rotation matrix , are double eigenvalues if and only if Q = rI for some real scalar valued integrable function r.