We give a general framework for constructing supersymmetric
solutions in the presence of non-trivial fluxes of tensor gauge fields.
This technique involves making a general Ansatz for the metric and then
defining the Killing spinors in terms of very simple projectors on the spinor
fields. These projectors and, through them, the spinors, are determined
algebraically in terms of the metric Ansatz. The Killing spinor equations
then fix the tensor gauge fields algebraically, and, with the Bianchi identities,
provide a system of equations for all the metric functions.
We illustrate this by constructing an infinite family of massive flows that
preserve eight supersymmetries in M-theory. This family constitutes all the
radially symmetric Coulomb branch flows of the softly broken, large N scalar-fermion
theory on M2-branes. We reduce the problem to the solution of a single,
non-linear partial differential equation in two variables. This
equation governs the flow of the fermion mass, and the function that
solves it then generates the entire M-theory solution algebraically in terms of
the function and its first derivatives.
While the governing equation is non-linear, it has a very simple perturbation theory
from which one can see how the Coulomb branch is encoded.