Vacua scan of $5d$, $N=2$ consistent truncations

In this letter we present a scan for new vacua within consistent truncations of eleven/ten-dimensional supergravity down to five dimensions that preserve $N=2$ supersymmetry, after their complete classification in arXiv:2112.03931. We first make explicit the link between the equations of exceptional Sasaki-Einstein backgrounds in arXiv:1602.02158 and the standard BPS equations for $5d$ $N=2$ of arXiv:1601.00482. This derivation allows us to expedite a scan for vacua preserving $N=2$ supersymmetry within the framework used for the classification presented in arXiv:2112.03931.