How to solve a n-dimensional matrix equation involving a matrix and its transpose?

Question

Example:

$A A^T = X$, where, for example, $X$ and $A$ are $N\times N$ matrices.

Typically you can solve it with plugging in variables and ending up with a system of equations with the same number of variables as equations. However, how can this be generalized to n-dimensional matrices?

$A A^T$ is symmetric.