  We establish analytically that the potential of N=8 supergravity in four
dimensions has a new N=1 supersymmetric critical point with U(1) x U(1)
symmetry. We work within a consistent N=1 supersymmetric truncation and obtain
the holographic flow to this new point from the N=8 critical point. The
operators that drive the flow in the dual field theory are identified and it is
suggested the new critical point might represent a new conformal phase in the
holographic fermion droplet models with sixteen supersymmetries. The flow also
has c_{IR}/c_{UV}} = 1/2. We examine the stability of all twenty known critical
points and show that the SO(3) x SO(3) point is a perturbatively stable
non-supersymmetric fixed point. We also locate and describe a novel critical
point that also has SO(3)\times SO(3) symmetry and is related to the previously
known one by triality in a similar manner to the way that the SO(7)^\pm points
are related to one another.
