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An algebraic Riccati equation can be changed into a Moser-Veselov equation by deriving the symmetric and skew-symmetric conditions from the algebraic Riccati equation. The symmetric condition is changed into orthonormal one by finding an appropriate transformation matrix. Ultimately, a closed-form solution of a particular case of algebraic Riccati equation is suggested by using an orthonormal solution of the Moser-Veselov equation. An application example is also included to demonstrate the effectiveness of the method proposed.