I want to find all matrices $G \in SO(3)$ that do not change the first row of elements in $SO(3)$ when right multiplying by $G$, i.e.
$$ \{ G \in SO(3): \forall A \in SO(3) \quad A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}\;, \\ \;\text{if} \; A' = A G \;\text{then}\; a_{11}' = a_{11}, a_{12}' = a_{12}, a_{13}' = a_{13} \} $$
I found that the matrices $G$ for having this with left-multiplication are precisely the rotations
$$ \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \alpha & -\sin \alpha \\ 0 & \sin \alpha & \cos \alpha \end{pmatrix} $$ around the first coordinate axis, but I can't seem to find a good characterization for the matrices on the right.
Hint: consider $A\in\left\{I,\ \pmatrix{0&1&0\\ 0&0&1\\ 1&0&0},\ \pmatrix{0&0&1\\ 1&0&0\\ 0&1&0}\right\}$.