Shear-free relativistic fluids and the absence of movable branch points

The problem of determining the metric for a non-static shear-free spherically symmetric fluid (either charged or neutral) reduces to the problem of deter- mining a one parameter family of solutions to a second-order ODE containing two arbitrary functions f and g. Choices for f and g are determined such that this ODE admits a one-parameter family of solutions that have poles as their only movable singularities. This property is strictly weaker than the Painleve property and it is used to identify classes of solvable models. It is shown that this procedure systematically generates many exact solutions in- cluding the Vaidya metric, which does not arise from the standard Painleve analysis of the second-order ODE. Interior solutions are matched to exte- rior Reissner-Nordstrøm metrics. Some solutions given in terms of second Painleve transcendents are described.