  We use the techniques of "algebraic Killing spinors" to obtain a family of
holographic flow solutions with four supersymmetries in M-theory. The family of
supersymmetric backgrounds constructed here includes the non-trivial flow to
the (2+1)-dimensional analog of the Leigh-Strassler fixed point as well as
generalizations that involve the M2-branes spreading in a radially symmetric
fashion on the Coulomb branch of this non-trivial fixed point theory. In
spreading out, these M2-branes also appear to undergo dielectric polarization
into M5-branes. Our results naturally extend the earlier applications of the
"algebraic Killing spinor" method and also generalize the harmonic Ansatz in
that our entire family of new supersymmetric backgrounds is characterized by
the solutions of a single, second-order, non-linear PDE. We also show that our
solution is a natural hybrid of special holonomy and the "dielectric
deformation" of the canonical supersymmetry projector on the M2-branes.
