  In the initial conditions of the $3 + 1$ formalism for numerical relativity,
the transverse and trace-free (TT) part of the extrinsic curvature plays a key
role. We know that TT tensors possess two degrees of freedom per space point.
However, finding an expression for a TT tensor depending on only two scalar
functions is a non-trivial task. Assuming either axial or translational
symmetry, expressions depending on two scalar potentials alone are derived here
for \emph{all} TT tensors in flat $3$-space. In a more general spatial slice,
only one of these potentials is found, the same potential given in
\cite{BakerPuzio} and \cite{Dain}, with the remaining equations reduced to a
partial differential equation, depending on boundary conditions for a solution.
As an exercise, we also derive the potentials which give the Bowen-York
curvature tensor in flat space.
