Solution of Lur´e equations via deflating subspaces

We consider the so-called Lur´e matrix equations that arise e.g. in linear-quadratic infinite time horizon optimal control. We characterize the set of solutions in terms of deflating subspaces of even matrix pencils. In particular, it is shown that there exist solutions which are extremal in terms of definiteness. It is shown how these special solutions can be constructed deflating subspaces of even matrix pencils.