In this note, the celebrated positive real lemma equations are considered with the purpose of obtaining necessary and sufficient conditions for the existence of solutions under the mildest system-theoretic assumptions. More precisely, necessary and sufficient conditions are established under the very weak assumption of sign-controllability. It is then shown that the order of the equations may be suitably reduced by restricting attention to a subspace related to a certain observable subsystem. This reduction, beside being interesting per se, permits us to weaken the assumption of sign-controllability. Finally, the assumptions are further weakened as to include the case when the state matrix has uncontrollable eigenvalues on the imaginary axis.