This is actually a homework problem.
The inverse boson operators $a^{-1}$ and $\left(a^\dagger\right)^{-1}$ are defined as $a^{-1} |n\rangle = \frac{1}{\sqrt{n+1}} |n+1\rangle$ $\left(a^\dagger\right)^{-1} |n\rangle = \begin{cases} \frac{1}{\sqrt{n}} |n-1\rangle & n>0 \\ 0 & n=0 \end{cases} $
$a^{-1}$ is in fact only right inverse of boson annihilation operator, and $\left(a^\dagger\right)^{-1}$ the left inverse of $\left(a^\dagger\right)$.
The question is how to construct a realization of Lie algebra $\mathfrak{so}(3)$ analogous to two boson realization of $\mathfrak{so}(3)$.