  We derive explicit reconstruction formulas for the attenuated geodesic X-ray
transform over functions and, in the case of non-vanishing attenuation, vector
fields, on a class of simple Riemannian surfaces with boundary. These formulas
partly rely on new explicit approaches to construct continuous right-inverses
for backprojection operators (and, in turn, holomorphic integrating factors),
which were previously unavailable in a systematic form. The reconstruction of
functions is presented in two ways, the latter one being motivated by numerical
considerations and successfully implemented at the end. Constructing the
right-inverses mentioned require that certain Fredholm equations, first
appearing in [Pestov-Uhlmann, IMRN 2004], be invertible. Whether this last
condition reduces the applicability of the overall approach to a strict subset
of simple surfaces remains open at present.
