Families of solutions of algebraic Riccati equations

We consider Homogeneous Algebraic Riccati Equations in the general situation when the matrix of the dynamics can be mixed. We show that in this case the equation may have infinitely many families of solutions. An analysis of these families is carried over and an explicit formula is derived. When the matrix of the dynamics is unmixed, this formula provides an explicit parametrization of the whole set of the solutions. We finally show that there are situations in which the equation has spurious solutions that do not belong to any of the families and derive sufficient conditions under which, on the contrary, the union of the families covers the whole set of solutions.