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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 thePestov-Uhlmann range characterization of the geodesic X-ray transform, byconstructing a basis of functions making up its range and co-kernel. Such arange characterization also translates into moment conditions {\it \`a la}Helgason-Ludwig or Gel'fand-Graev. We then derive an explicit Singular ValueDecomposition for the geodesic X-ray transform. Computations dictate a specificchoice 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 $\kappa\in (-1,1)$.

Range characterizations and Singular Value Decomposition of the geodesic X-ray transform on disks of constant curvature