Steadily rotating perfect-fluid gravitating prolate spheroids in Weyl gravity

It is shown that the equations of hydrodynamics do not admit solutions describing gravitating, incompressible, perfect fluid, prolate spheroids within the context of Weyl gravity, i.e. a theory whose static, spherically symmetric solution admits, besides the usual 1/r potential, an additional linear one, which has been invoked to solve the galactic rotation curve problem without the introduction of `dark matter'. This extends the similar, recent result obtained by Florides and Spyrou within Newtonian gravity to this alternative gravitational theory.