  We consider 3D flow equations inspired by the renormalization group (RG)
equations of string theory with a three dimensional target space. By modifying
the flow equations to include a U(1) gauge field, and adding carefully chosen
De Turck terms, we are able to extend recent 2D results of Bakas to the case of
a 3D Riemannian metric with one Killing vector. In particular, we show that the
RG flow with De Turck terms can be reduced to two equations: the continual Toda
flow solved by Bakas, plus its linearizaton. We find exact solutions which flow
to homogeneous but not always isotropic geometries.
