  In this article, we construct a class of explicit, smooth and spherically
symmetric solutions to the asymptotically flat vacuum constraint equations
which have ADM mass of arbitrary sign ($- \infty$, negative, zero, positive).
As a direct consequence, there exist asymptotically flat vacuum initial data
sets whose metrics are exactly negative mass Schwarzschild outside a given
ball. We emphasize that our result does not contradict the spacetime positive
energy theorem proven by Eichmair, instead it shows that the decay rate at
infinity of the symmetric $(0,2)$-tensor $k$ stated in the theorem is sharp.
The key argument we use in the article is classical, based on the conformal
method, in which the conformal equations are equivalently transformed into a
single nonlinear equation of functions of one variable.
