  In teleparallel geometries, symmetries are represented by affine frame
symmetries which constrain both the (co)frame basis and the spin-connection
(which are the primary geometric objects). In this paper we shall study
teleparallel geometries with a single affine symmetry, utilizing the locally
Lorentz covariant approach and adopting a complex null gauge. We first
introduce an algorithm to study geometries with an affine frame symmetry, which
consists of choosing coordinates adapted to the symmetry, constructing a
canonical frame, and solving the equations describing the symmetry. %for the
spin connection (to determine the six real Lorentz functions. All of the
constraints on the geometry are determined in the case of a single affine
symmetry, but there are additional constraints arising from the field equations
for a given theory of teleparallel gravity. In particular, we find that in
$f(T)$ teleparallel gravity there will be severe constraints on the geometry
arising from the antisymmetric part of the field equations.
