I am looking for the value of $\mathbf{X}$ in a function of the type
\begin{align} (\mathbf{X}-\mathbf{A})e^{\mathbf{X}}e^{-\mathbf{A}} = \mathbf{B} \end{align}
where $\mathbf{A}$,$\mathbf{B}$,$\mathbf{X}\in \Re^{n\times n}$. I assume that $\mathbf{A}$ and $\mathbf{X}$ do not commute, i.e., $\mathbf{A}\cdot\mathbf{X} \neq \mathbf{A}\cdot\mathbf{X}$ which means that $e^{\mathbf{X}}e^{-\mathbf{A}} \neq e^{\mathbf{X}-\mathbf{A}}$. This problem is similar to the Lambert W function but in a more generic setting.
I am trying to find an analytical solution which gives all the infinite possible values in a similar manner to the Lambert W function.
Does anyone know if this problem has been solved or has any hint on how to approach it?