  Let $(M,J,\Omega)$ be a closed polarized complex manifold of K\"ahler type.
Let $G$ be the maximal compact subgroup of the automorphism group of $(M,J)$.
On the space of K\"ahler metrics that are invariant under $G$ and represent the
cohomology class $\Omega$, we define a flow equation whose critical points are
extremal metrics, those that minimize the square of the $L^2$-norm of the
scalar curvature. We prove that the dynamical system in this space of metrics
defined by the said flow does not have periodic orbits, and that its only fixed
points are extremal metrics. We prove local time existence of the flow, and
conclude that if the lifespan of the solution is finite, then the supremum of
the norm of its curvature tensor must blow-up as time approaches it. While
doing this, we also prove that extremal solitons can only exist in the
non-compact case, and that the range of the holomorphy potential of the scalar
curvature is an interval independent of the metric chosen to represent
$\Omega$. We end up with some conjectures concerning the plausible existence
and convergence of global solutions under suitable geometric conditions.
