I want to solve the following matrix equation with respect to the matrix variable $\mathbf{X}$ which is a real symmetric positive definite matrix. The given matrix $\mathbf{A}$ is real and symmetric, "a" is a scalar, and $\mathbf{I}$ is the identity matrix of the appropriate size.
$$\mathbf{X}^{-1}\mathbf{X}^{-1} + \big(\mathbf{X}-\mathbf{I}\big)^{-1} +a\mathbf{X}= \mathbf{A} $$
If we ignore the first term in this equation, there is a closed form solution in the following paper, eq. (14) and Lemma 2.1 (I need a wise closed form solution like this one!) http://www.optimization-online.org/DB_FILE/2009/09/2409.pdf
Hint: Try to show that if $A$ has the eigendecomposition $V\Lambda V^T$ then $X$ must have the eigendecomposition $VDV^T$ where $\Lambda = D^{-2}+(D-I)^{-1}-aD$. In other words, $X$ must have the same eigenvectors as $A$ and for each eigenpair $(\lambda,v)$ of $A$, we have that $(d,v)$ is an eigenpair of $D$ where $\lambda = \dfrac{1}{d^2}+\dfrac{1}{d-1}-ad$.
From there, you simply need to solve the equation $\lambda = \dfrac{1}{d^2}+\dfrac{1}{d-1}-ad$ for $d$ in terms of each eigenvalue $\lambda$ of $A$.