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In the present paper we obtain a closed-form solution for the class of continuous-time algebraic Riccati equations (AREs) with vanishing state weight. The AREs in such a class solve a minimum energy control problem. The present work extends on previous preliminary results by considering the more challenging case of repeated unstable eigenvalues. The obtained closed-form solution gives insight on issues such as loss of controllability and it might also prove comparable in terms of numerical precision over current solving algorithms.