Given a symmetric and positive definite matrix $B$, solve the following matrix equation for matrix $X$ $$ X B - B^{-1} \frac{\mathrm{tr} \left( B^{-3} X B \right)}{\mathrm{tr} \left( B^{-4} \right)} = 0 $$
I tried it with the strategies from Solving matrix equation involving trace and Solving for a matrix in an equation with trace, but for the given equation, this only leads to trivial results such as
$$\mathrm{tr}(B^{-3} X B)= \mathrm{tr}(B^{-3} X B)$$
Additionally, is there even a general strategy to solve this type of equations? I can only stare at the equation and hope for some insight right now. Is there even a solution? Or are there infinitely many?