In this paper, we first consider the
𝕊
ℙ
𝔽
{\mathbb{SPF}}
equation on a Riemannian CRVF-manifold M and show that either M is Einstein or the potential function is pointwise collinear with ζ on an open set U of M. Next, we show that if a Riemannian CRVF-manifold M is the spatial factor of a
𝕊
ℙ
𝔽
{\mathbb{SPF}}
with a Batch tensor then it is a Batch flat space-time manifold.