Given symmetric $A,B\in\Bbb R^{n\times n}$ what is a good description of collection of $X\in\Bbb R^{n\times n}$ such that $$XX'=I$$ $$AX=XB$$ 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 $UA=AU$ or $BU=UB$?

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

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 $\Bbb R$?

Given symmetric $A,B\in\Bbb R^{n\times n}$ what is a good description of collection of $X\in\Bbb R^{n\times n}$ such that $$XX'=I$$ $$AX=XB$$ holds?

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

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

It is easy to see if we find one such $X$ we can ALL such $X$ from finding $U\in\Bbb R^{n\times n} $ such that $UU'=I$ and $UA=AU$ or $BU=UB$.

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

Given symmetric $A,B\in\Bbb R^{n\times n}$ what is a good description of collection of $X\in\Bbb R^{n\times n}$ such that $$XX'=I$$ $$AX=XB$$ 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 $UA=AU$ or $BU=UB$?

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

Given symmetric $A,B\in\Bbb R^{n\times n}$ what is a good description of collection of $X\in\Bbb R^{n\times n}$ such that $$XX'=I$$ $$AX=XB$$ holds?

(1) What if $A,B,X\in\Bbb F_{p^r}^{n\times n}$ is considered?

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

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

(3) Suppose we know one such $X$ we can find all such $X$ from finding $U$ such that $UU'=I$ and $UA=AU$ or $BU=UB$. Do all such $X$ arise this way?

Given symmetric $A,B\in\Bbb R^{n\times n}$ what is a good description of collection of $X\in\Bbb R^{n\times n}$ such that $$XX'=I$$ $$AX=XB$$ holds?

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

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

It is easy to see if we find one such $X$ we can ALL such $X$ from finding $U\in\Bbb R^{n\times n} $ such that $UU'=I$ and $UA=AU$ or $BU=UB$.

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