Consider the following quantity $$X^T (XX^T + \mathrm{Id})^{-1} X,$$ where $X \in \mathbb{R}^{m\times n}$ is a iid random matrix with 0 mean and finite variance.

The empiric covariance matrix ${X^T X}$ has been extensively studied and the asymptotic (when $n,m\rightarrow \infty$ and $n/m\rightarrow \gamma$) law of its eigenvalues is for instance knwon as the Marchenko-Pastur distribution after adequate renormalizations.

Is there a generalization or similar analyses for the case of the matrix introduced above, which naturally arise in the context of nested linear models as a modification of $X^T X$?