Let $A$ be a $2n\times 2n$ real matrix with the following structure \begin{equation} A = \left(\begin{matrix} 0 & -I \\ K & S \end{matrix}\right) \end{equation} with all sub-matrices of size $n\times n$: $I$ is the identity matrix, $K$ is symmetric positive definite and $S$ is diagonal but singular. I am interested in the (numerical) solution of the continuous time Lyapunov equation for the $2n\times 2n$ matrix R: \begin{equation} R A^\text{T} + A R = \left(\begin{matrix} 0 & 0 \\ 0 & \Gamma \end{matrix}\right) \end{equation} where $\Gamma$ is a diagonal and singular $n\times n$ real matrix.
However, I only need a few elements of $R$. More specifically, writing
\begin{equation} R = \left(\begin{matrix} X & C \\ C^\text{T} & V \end{matrix}\right) \end{equation} all I really want are the diagonal entries of $V$ (also $n\times n$).
Is there anyway I could reduce this problem to some other (probably Sylvester) equation for $V$ or, better yet, only it's diagonal? I don't really know how to approach this problem.