Say we have a matrix $A\in\mathbb{R}^{n\times n}$ that can be written as $A=(M+\lambda I\_n)^\top (M-\lambda I\_n)$, where $M\in\mathbb{R}^{n\times n}$ and $\lambda\in\mathbb{R}\backslash\{0\}$.

I am wondering for any vector $\xi\in\mathbb{R}^n$, whether the following linear system

$A^\top(A^2+A)X=A^\top\xi$

may have no solution. Many thanks.