Gauge invariance and Gauss's law

Question

I am reading an article: Stable Gapless Bose Liquid Phases without Any Symmetry (and also see Pretko's paper on the same subject). There, the authors wrote that Gauss's laws are associated with different gauge transformations as summarized in table 1 of Ref. 1. I am not quite sure how the authors obtained the Gauss's laws only from the gauge transformation of the vector potential. Take the simplest case rank-1 (the classical E&M) theory for example. The gauge transformation

\[A_i \to A_i + \partial_i \lambda\]

where $\lambda$ is a scalar function of spacetime is accompanied by a corresponding change in the electrostatic potential $\phi \to \phi - \partial_t \lambda$ for the electric field to be invariant. How does this imply Gauss's law $\partial_i E^i =\nabla \cdot \vec{E} =0$? Are there some hidden assumptions? Could someone please help?

[cross posted on SE: https://physics.stackexchange.com/questions/597398/gauge-invariance-and-gausss-law]