Given the real matrices $A$ and $B = B^T$, how can I show that there exists a real matrix $C$ such that $AC + C^T A^T + C^T C \leq B$ if and only if $B + AA^T \geq 0$?
For the "only if" part, what I tried is as follows: Because $AC + C^T A^T + C^T C \leq B$, then there exists a positive semi-definite matrix $Q$ such that $AC + C^T A^T + C^T C + Q = B$. This almost looks like an algebraic Riccati equation in $C$, except the $C^T A^T$ term should be $CA^T$. I'm not sure how to proceed from here.
If $B+AA^T\ge0$, then $AC+C^TA^T+C^TC\le B$ when $C=-A^T$.
Conversely, if $AC+C^TA^T+C^TC\le B$, then $$ B+AA^T\ge AC+C^TA^T+C^TC+AA^T=(A+C^T)(A+C^T)^T\ge0. $$