Not equal to this (my) own question.
It's more general, probably more easy than the original question.
All of the elements of $X$ and $A$ are integers.
$XX^\top=A$ and $A$ is a symmetric matrix. How to find all possible $X$ matrices?
Maybe a Gram-Schmidt method to keep only integer solutions.
An example:
XX^\top= \left( \begin{array}{ccc} 0 & 1 & 1 \ 1 & 0 & 1 \ 1 & 1 & 0 \end{array} \right)
\left( \begin{array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}
\right)
\left( \begin{array}{ccc} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{array} \right)=A