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The authors give the exact spherically symmetric solution of the Einstein field equations for dust (p=0) in observational coordinates for data on the past light cone C-(p0) consisting of redshifts, observer area distances, and galaxy number counts. A Bondi potential is found, which facilitates the integration. Also they show in detail how FLRW models can be constructed via these solutions from FLRW data. Finally, they discuss their approach with respect to other observationally oriented treatments, particularly those of Ehlers and Rindler (1987), and Rindler and Suson (1989).

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The authors describe the fluid-ray (FR) tetrad, which is based on the four-velocity u of the cosmological fluid and the null vector k which lies along the generators of the null geodesics in the past light cones C-(p0) centred on the worldline C. Then, they formulate the field equations in terms of this tetrad, giving also the Jacobi identities, the metric variable equations for observational coordinates, and the contracted Bianchi identities. The equivalent set of equations, involving the Ricci identities, the full Bianchi identities, along with the contracted Bianchi identities, is also given. This formulation facilitates the integration of the field equations in certain cosmological contexts.

The integrability conditions for a certain second order ordinary differential equation in two variables are studied via the concept of decomposability of the associated differential operator. The results are applied to regain in a unified manner the known exact solutions for locally rotationally symmetric, spatially homogeneous cosmological models. In addition, new solutions are obtained.