While investing an optimization problem, I ended up with the following quadratic matrix equation $$ P^T Q P - A^T P - P^T A + R = 0$$ in which $Q, A$ and $R$ are $n\times n$ real matrices and $Q$ and $R$ are symmetric positive. $P$ is the unknown matrix to be determined.
Of course, if we consider $P = P^T$, we recover the standard Ricatti equation for which conditions for the existence and potential unicity of symmetric solutions are well-known to me. Unfortunately, I do not manage to find any reference on the above "non-symmetric" equivalent.
Is any of you aware of references or results on the existence of solutions to this equation?
Thanks in advance for your help !