  The transverse and trace-free (TT) part of the extrinsic curvature represents
half of the dynamical degrees of freedom of the gravitational field in the $3 +
1$ formalism. As such, it is part of the freely specifiable initial data for
numerical relativity. Though TT tensors in 3-space possess only two component
degrees of freedom, they cannot ordinarily be given solely by two scalar
potentials. Such expressions have been derived however, in coordinate form, for
all TT tensors in flat space which are also translationally or axially
symmetric (Conboye and \'O Murchadha 2014 \emph{Class. Quantum Grav.} {\bf 31}
085019). Since TT tensors are conformally covariant, these also give TT tensors
in conformally-flat space. In this article, the work above has been extended by
giving a coordinate-\emph{independent} expression for these TT tensors. The
translational and axial symmetry conditions have also been generalized to
invariance along any hypersurface orthogonal Killing vector.
