The moduli space of null Sasaki $η$-Einstein structures for simply connected compact 5-dimensional manifold $M$ is determined by the following quadric
$$\{[\alpha]\in H^2(M,\mathbb C) \; \text{such that} \; |([\alpha],[\alpha])|+|([\alpha],[\bar\alpha])|>0\}/\mathbb C^*$$
Is there any explicit moduli space of null Sasaki $η$-Einstein structures for higher dimensions?