The equationy´´ =f(x)y2arises in the study of a class of fluid models in relativity and possesses the Painlevéproperty (closely connected with integrability) if and only iffsatisfies a certain sixth-order ODE which admitsGL(2,) as its symmetry group. Using differential invariants of this non-solvable group, the general solution is obtained. A special case of the sixth-order equation is equivalent to the generalized Chazy equation with parametern= (6/7). All known explicit choices forfconsidered in the literature arise in a natural way in this framework. Generalizations of the techniques described here lead to a novel class of integrable equatio
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