On the existence of full set of solutions of Riccati equations

We consider the algebraic Riccati equations in general case of arbitrary pair (A, B) and nonsingular matrix Γ of quadratic form. We obtain the necessary and sufficient conditions for existence of a complete set of solutions. Thus, even in the standard case of sign definite matrix Γ we maximally weaken the restrictions on pair (A, B). These results are extended to the case of singular matrix Γ. We derive and use a special representation of Hamiltonian matrices which form a basis for proofs.