We shall consider the matrix-valued differential operator

$$(L u)(x) :=u'(x) - \begin{pmatrix} 0 & \sin(2\pi x-\frac{\pi}{6})\\ - 2\sin(2\pi x+\frac{\pi}{6}) & 0 \end{pmatrix} u(x).$$

This is a $1$-periodic operator. Thus, does there exist a $\lambda \in \mathbb C$ and a $1$-periodic solution to this ODE such that

$$ (L - \lambda)u = 0.$$

Probably there is no explicit solution, but can we show the existence of such a solution?