Let $P$ and $Q$ be positive definite matrices. Consider the following matrix equation
$$\label{star}\tag{$\star$}
XPX^\top - P = -Q, \quad X\in\mathbb{R}^{n\times n}.
$$
Moreover, given a matrix $M\in\mathbb{R}^{n\times n}$ let $\sigma(M)$ denote the spectrum of $M$ and $\lambda_{\min}(M):=\min_{\lambda\in\sigma(M)} |\lambda| $, $\lambda_{\max}(M):=\max_{\lambda\in\sigma(M)} |\lambda|$.  


> 1. Is it true that any solution of \eqref{star} can be written as $X=(P-Q)^{1/2}TP^{-1/2}$ with $T$ being an arbitrary orthogonal matrix?

> 2. Is it true that the eigenvalues of any $X$ solving \eqref{star} have moduli contained in the interval $[\lambda_{\min}((P-Q)^{1/2}P^{-1/2}),\lambda_{\max}((P-Q)^{1/2}P^{-1/2})]$?