How do you solve for $X \in \mathbb{R}^{m \times n}$ given an equation of the form
$$(kX + A)^T \: (kX + A) = B_k$$
for a given $k \in \mathbb{R}$, $A \in \mathbb{R}^{m \times n}$ and a symmetric matrix $B_k \in \mathbb{R}^{n \times n}$.
I suppose you can always define $Z \triangleq kX + A$, such that
$$Z^T Z = B_k$$
This allows for a solution, $Z = \Lambda^{1/2} \: U^T$ where $\Lambda$ and $U$ are the respective eigenvalues and eigenvectors of $B_k$. However, such a solution is not unique. So, I cannot solve for $X$ as $\dfrac{1}{k} \left( Z - A \right)$.
Q1: Is this line of reasoning correct?
Q2: Can a unique solution be derived if $B_k$ is available for multiple values of $k$?
I have also tried to use a combination of vectorization and Kronecker product to write it into a more recognizable form but haven't managed to do so. Even if a unique solution is not possible, I would like to get to know under what conditions on $X$, $A$ and $B_k$ are such type of equations solvable? A reference to a mathematical source would be well appreciated!