  For a one-parameter family of simple metrics of constant curvature ($4\kappa$
for $\kappa\in (-1,1)$) on the unit disk $M$, we first make explicit the
Pestov-Uhlmann range characterization of the geodesic X-ray transform, by
constructing a basis of functions making up its range and co-kernel. Such a
range characterization also translates into moment conditions {\it \`a la}
Helgason-Ludwig or Gel'fand-Graev. We then derive an explicit Singular Value
Decomposition for the geodesic X-ray transform. Computations dictate a specific
choice of weighted $L^2-L^2$ setting which is equivalent to the $L^2(M,
dVol_\kappa)\to L^2(\partial_+ SM, d\Sigma^2)$ one for any $\kappa\in (-1,1)$.
