  There is a common description of different intrinsic geometric flows in two
dimensions using Toda field equations associated to continual Lie algebras that
incorporate the deformation variable t into their system. The Ricci flow admits
zero curvature formulation in terms of an infinite dimensional algebra with
Cartan operator d/dt. Likewise, the Calabi flow arises as Toda field equation
associated to a supercontinual algebra with odd Cartan operator d/d \theta -
\theta d/dt. Thus, taking the square root of the Cartan operator allows to
connect the two distinct classes of geometric deformations of second and fourth
order, respectively. The algebra is also used to construct formal solutions of
the Calabi flow in terms of free fields by Backlund transformations, as for the
Ricci flow. Some applications of the present framework to the general class of
Robinson-Trautman metrics that describe spherical gravitational radiation in
vacuum in four space-time dimensions are also discussed. Further iteration of
the algorithm allows to construct an infinite hierarchy of higher order
geometric flows, which are integrable in two dimensions and they admit
immediate generalization to Kahler manifolds in all dimensions. These flows
provide examples of more general deformations introduced by Calabi that
preserve the Kahler class and minimize the quadratic curvature functional for
extremal metrics.
