  We introduce a new concept of unbounded solutions to the operator Riccati
equation $A_1 X - X A_0 - X V X + V^\ast = 0$ and give a complete description
of its solutions associated with the spectral graph subspaces of the block
operator matrix $\mathbf{B} = \begin{pmatrix} A_0 & V V^\ast & A_1
\end{pmatrix}$. We also provide a new characterization of the set of all
contractive solutions under the assumption that the Riccati equation has a
contractive solution associated with a spectral subspace of the operator
$\mathbf{B}$. In this case we establish a criterion for the uniqueness of
contractive solutions.
