  We continue our previous analysis (hep-th/0412045) of 1/2 BPS solutions to
minimal 6d supergravity of bubbling form. We show that, by turning on an axion
field in the T^2 torus reduction, the constraint F \wedge F, present in the
case of an S^1 x S^1 reduction, is relaxed. We prove that the four-dimensional
reduction to a bosonic field theory, whose content is the metric, a gauge
field, two scalars and a pseudo-scalar (the axion), is consistent. Moreover,
these reductions when lifted to the six-dimensional minimal supergravity
represent the sought-after family of 1/2 BPS bubbling solutions.
