  We explore the family of fixed points of T-Duality transformations in three
dimensions. For the simplest nontrivial self-duality conditions it is possible
to show that, additionally to the spacelike isometry in which the T-Duality
transformation is performed, these backgrounds must be necessarily stationary.
This allows to prove that for nontrivial string coupling, the low energy
bosonic string backgrounds which are additionally self-T-dual along an isometry
direction generated by a constant norm Killing vector are uniquely described by
a two-parametric class, including only three nonsingular cases: the charged
black string, the exact gravitational wave propagating along the extremal black
string, and the flat space with a linear dilaton. Besides, for constant string
coupling, the only self-T-dual lower energy string background under the same
assumptions corresponds to the Coussaert-Henneaux spacetime. Thus, we identify
minimum criteria that yield a classification of these quoted examples and only
these. All these T-dual fixed points describe exact backgrounds of string
theory.
