An explicit nonempty sourced family

The metric coefficient in (eq:jet2) is an unknown, not an external tensor. We now solve a nontrivial subclass.

For compact smooth spatial $b_0$, prescribe the auxiliary flat Newton branch

The kernel is locally integrable, and derivatives can be transferred to the smooth compact source. This gives a smooth potential. The identity follows from the flux of $\nabla (1/|x|)$ through a small sphere.

This initial-data branch is additional model data. It is not an intrinsic conical inverse.