A first-order boundary value problem with boundary condition on a countable set of points

Let $E=\{E_n\}$ be the family of subspaces spanning the eigenfunctions and adjoint functions of the boundary-value problem
$$ -i\frac{dy}{dx}=\lambda y,\quad -a\le x\le a,\qquad U(y)\equiv\int_{-a}^ay(t)d\sigma(t)=0, $$
that correspond to “close” eigenvalues (in the sense of the distance defined as the maximal of the Euclidean and the hyperbolic metrics). For a purely discrete measure $d\sigma$ it is shown that the system $E$ does not form an unconditional basis of subspaces in $L^2(-a,a)$ if at least one of the end points $\pm a$ is mass-free.