We know that SO(3) matrix stands for the proper rotation in 3D space. But when I read this paper, there is a SO(3) matrix stands for the general query matrix of Grover's algorithm in SO(3) form: $$ \left(\begin{array}{ccc} R_{11} & R_{12} & R_{13} \\ R_{21} & R_{22} & R_{23} \\ R_{31} & R_{32} & R_{33} \end{array}\right), $$ where $$R_{11}=\cos \phi\left(\cos ^{2} 2 \beta \cos \theta+\sin ^{2} 2 \beta\right)+\cos 2 \beta \sin \theta \sin \phi\\R_{12}=\cos 2 \beta \cos \phi \sin \theta- \cos \theta \sin \phi\\R_{13}=-\cos \phi \sin 4 \beta \sin ^{2} \frac{\theta}{2}+\sin 2 \beta \sin \theta \sin \phi\\R_{21}=-\cos (2 \beta) \cos \phi \sin \theta+ \left(\cos ^{2} \frac{\theta}{2}-\cos 4 \beta \sin ^{2} \frac{\theta}{2}\right) \sin \phi\\R_{22}=\cos \theta \cos \phi+\cos 2 \beta \sin \theta \sin \phi\\ R_{23}=-\cos \phi \sin 2 \beta \sin \theta- \sin 4 \beta \sin ^{2} \frac{\theta}{2} \sin \phi\\R_{31}=-\sin 4 \beta \sin ^{2} \frac{\theta}{2}\\R_{32}=\sin 2 \beta \sin \theta\\R_{33}=\cos ^{2} 2 \beta+\cos \theta \sin ^{2} 2 \beta.$$ The paper says that the eigenvector of this matrix is $\mathbf{l}=\left(\cot \frac{\phi}{2},1,-\cot 2 \beta \cot \frac{\phi}{2}+\cot \frac{\theta}{2} \csc 2 \beta\right)^{T}$.
I know this question is very basic and I've tried to use Matlab to calculate it. But I just can't figure out how can the author got the eigenvector of such a simple form? Can it be manually calculated? Is there a better way to compute the eigenvector of this kind of parameterized matrix?