  The integrability properties of the field equation $L_{xx} = F(x)L^2$ of a
spherically symmetric shear--free fluid are investigated. A first integral,
subject to an integrability condition on $F(x)$, is found, giving a new class
of solutions which contains the solutions of Stephani (1983) and Srivastava
(1987) as special cases. The integrability condition on $F(x)$ is reduced to a
quadrature which is expressible in terms of elliptic integrals in general.
There are three classes of solution and in general the solution of $L_{xx} =
F(x)L^2$ can only be written in parametric form. The case for which $F=F(x)$
can be explicitly given corresponds to the solution of Stephani (1983). A Lie
analysis of $L_{xx} = F(x) L^2$ is also performed. If a constant $\alpha$
vanishes, then the solutions of Kustaanheimo and Qvist (1948) and of this paper
are regained. For $\alpha \neq 0$ we reduce the problem to a simpler,
autonomous equation. The applicability of the Painlev\'e analysis is also
briefly considered.
