On a transformation operator for a system of Sturm–Liouville equations

We prove the existence of a transformation operator with a condition at infinity that sends a solution of the matrix equation $-y''+My=\lambda^2y$ ($M$ is a constant Hermitian matrix) into a solution of the matrix equation $-y''+Q(x)y+My=\lambda^2y$ (the matrix function $Q(x)$ is continuously differentiable for $0\leqslant x<\infty$ and it is Hermitian for each $x$ belonging to $[0,\infty)$); we study some properties of the kernel of the transformation operator.