On the interplay between CPE metrics and $\sigma_2$-singular spaces

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 Eisntein in therms of $\sigma_2$-singular spaces introduced in \cite{silvaandrade2018}. Such a result improves our understanding about CPE metrics and Besse's conjecture (see \cite{besse2007einstein}) with a new geometric point of view.