Given a linear time-invariant control system with a quadratic performance criterion and output \(y=Cx\), it is known that the gain matrix of the optimal static feedback \(u=Fy\) can be obtained by solving a system of nonlinear matrix algebraic equations. A known iterative algorithm for solving this system requires at every step the solution of two Lyapunov- type matrix equations. The paper is devoted to a solution procedure for the latter equations, in the case where the original control system is singularly perturbed (a small parameter \(\epsilon\) multiplies a part of the derivatives). A complete slow-fast decomposition is achieved so that only low-order systems are involved in the computations at every step. The accuracy after k steps is proved to be \(O(\epsilon^ k)\).