Consider the following sequence of problem: With $A \in \mathbb{R}^{n \times m}$, $m>n$, and $x \in \mathbb{R}^m$, I am looking to linearly combine (non-trivially) the elements of the vector $Ax$ to zero, regardless of the value of $x$. Hence, I am looking for $y \in \mathbb{R}^n$ such that $y^TAx = 0$. This is relatively easy by choosing any $y \in \mathrm{Null}(A^T)$.
Now I extend this problem by saying that I want the same for a matrix. This matrix happens to be given by $A^T X A$, this time $X \in \mathbb{R}^{n \times n}$. Again, I want to find non-trivial linear combinations of the matrix elements such that their sum is zero. Again, this is not too difficult, since I can choose $y \in \mathrm{Null}(A)$ to find $y^T A^T X A x = 0$, for any $X$ and $x$.
Next, we consider the case of symmetric $X$. In this case, while $y \in \mathrm{Null}(A)$ is still sufficient, it may not be necessary. Consider $A = I_2$: then, $\mathrm{Null}(A) = \emptyset$, but I can still choose $1 \cdot X_{12} + (-1) \cdot X_{21} = 0$. However, I fail to see how I can arrive at this solution through $y^T X x$. Obviously, two vectors of dimension $n$ do not have enough degrees of freedom to encode all linear combinations of the $n^2$ matrix elements.
Finally, consider $X$ known, and assume $X = (2, 0; 1, 0)$, $A = I_2$. Again, $\mathrm{Null}(A)=\emptyset$, but we can find $x = (1,0)^T$ and $y = (0,1)^T$ such that $y^T A^T X A x = 0$.
So my question is, how can I express all solutions to the above problem in a closed form?