  Cotton flow tends to evolve a given initial metric on a three manifold to a
conformally flat one. Here we expound upon the earlier work on Cotton flow and
study the linearized version of it around a generic initial metric by employing
a modified form of the DeTurck trick. We show that the flow around the flat
space, as a critical point, reduces to an anisotropic generalization of
linearized KdV equation with complex dispersion relations one of which is an
unstable mode, rendering the flat space unstable under small perturbations. We
also show that Einstein spaces and some conformally flat non-Einstein spaces
are linearly unstable. We refine the gradient flow formalism and compute the
second variation of the entropy and show that generic critical points are
extended Cotton solitons. We study some properties of these solutions and find
a Topologically Massive soliton that is built from Cotton and Ricci solitons.
In the Lorentzian signature, we also show that the pp-wave metrics are both
Cotton and Ricci solitons.
