Basis property of eigenfunctions of a boundary value problem for a $2 \times 2$ system of ordinary differential equations

We consider the boundary value problem generated on the finite closed interval $x \in [0, 1]$ by the $2 \times 2$ system of ordinary differential equations
$$ y' - B(x)y=\lambda A(x)y, \qquad A(x)=\operatorname{diag}\{a_1(x), a_2(x)\}, \quad a_1(x) < 0 < a_2(x), $$
and the boundary conditions
$$ U_0y(0) + U_1y(1)=0, $$
where $y(x)=(y_1(x), y_2(x))^\top$, $U_0$ and $U_1$ are constant $(2 \times 2)$ matrices, and the system coefficients $a_j$ and $b_{jk}$ are assumed to be absolutely continuous. In the regular case, we prove that the system of eigenfunctions and associated functions forms a Schauder basis in the space $(L_p[0, 1])^2$, and in the case of almost regular boundary value problem of order $m \in \mathbb{N}$, we prove the Schauder basis property in some subspace of the space $(W_{p}^{m}[0, 1])^2$ with respect to the $(L_p[0, 1])^2$-norm.