  We call CPE metrics the critical points of the total scalar curvature
functional restricted to the space of metrics with constant scalar curvature of
unitary volume. In this short note, we give a necessary and sufficient
condition for a CPE metric to be Einstein in therms of $\sigma_2$-singular
spaces. Such a result improves our understanding about CPE metrics and Besse's
conjecture with a new geometric point of view. Moreover, we prove that the CPE
condition can be replaced by the related vacuum static space condition to
characterize closed Einstein manifolds in terms of $\sigma_2$-singular spaces.
