Given list of symmetric matrices $\{A_i,B_i\}_{i=1}^r\in\Bbb R^{n\times n}$ where $r\in\Bbb N$ is arbitrary what is a good description of collection of $X\in\Bbb R^{n\times n}$ such that $$XX'=I$$ $$A_iX=XB_i$$ holds?

Note: $X'$ means the transpose of $X$.

**(1)** Is there a test to see if there is no such $X$?

**(2)** Is it easy to see if we find one such $X$ we can find *ALL* such $X$ from finding $U\in\Bbb R^{n\times n} $ such that $UU'=I$ and $UA_i=A_iU$ or $B_iU=UB_i$ at every $i\in\{1,\dots,r\}$?

**(3)** What if we have $\Bbb F_{p^r}$ instead of $\Bbb R$?