Other Gaussian fields with stationary hyperuniform zero sets

Determine whether there exist Gaussian fields, beyond the planar Gaussian analytic function, whose zero sets are stationary hyperuniform point processes; specifically, construct such a Gaussian field (in the plane or higher dimensions) with a stationary hyperuniform zero set or prove that no such field exists.

Background

The planar Gaussian analytic function (GAF) has a zero set that is a stationary hyperuniform point process, a rare and highly structured example within Gaussian fields. For one-dimensional stationary fields, such a phenomenon is known to be impossible.

Identifying additional Gaussian fields with stationary hyperuniform zero sets (or proving their nonexistence) would clarify the scope of hyperuniformity within Gaussian nodal geometry and connect to broader cancellation phenomena for Gaussian random measures.