Negative-semidefinite solutions of the ARE R(X) = A*X + XA + XBB*X - C*C = 0 are studied. With respect to an appropriate basis die ARE breaks up into a Lyapunov equation A0*X0 + X0A0 = 0, where A0 has only purely imaginary eigenvalues, and an indecomposable Riccati equation R(r)(X(r)) = A(r)*X(r) + X(r)A(r) + X(r)B(r)B(r)*X(r) - C(r)*C(r) = 0 such that each solution X less-than-or-equal-to 0 is of the form X = diag(X0, X(r)). The focus is on the solutions S = {XX = diag(0, X(r)), R(r)(X(r)) = 0, X(r) less-than-or-equal-to 0}. The set S has as an order-isomorphic image a well-defined set N of A-invariant subspaces. The characterization of N involves the stabilizable and the uncontrollable subspace of (A, B, C).