  In the companion paper [SciPost Phys. 13, 108 (2022), arXiv:2205.11401
[hep-th]] we have studied the solution space at null infinity for gravity in
the partial Bondi gauge. This partial gauge enables to recover as particular
cases and among other choices the Bondi-Sachs and Newman-Unti gauges, and to
approach the question of the most general boundary conditions and asymptotic
charges in gravity. Here we compute and study the asymptotic charges and their
algebra in this partial Bondi gauge, by focusing on the flat case with a
varying boundary metric $\delta q_{AB}\neq0$. In addition to the
super-translations, super-rotations, and Weyl transformations, we find two
extra asymptotic symmetries associated with non-vanishing charges labelled by
free functions in the solution space. These new symmetries arise from a weaker
definition of the radial coordinate and switch on traces in the transverse
metric. We also exhibit complete gauge fixing conditions in which these extra
asymptotic symmetries and charges survive. As a byproduct of this calculation
we obtain the charges in Newman-Unti gauge, in which one of these extra
asymptotic charges is already non-vanishing. We also apply the formula for the
charges in the partial Bondi gauge to the computation of the charges for the
Kerr spacetime in Bondi coordinates.
