New rotation laws have been recently found for general-relativistic self-gravitating stationary fluids. It was not clear whether they apply to systems rotating with a constant linear velocity. In this paper we fill this gap. The answer is positive. That means, in particular, that these systems should exhibit the recently discovered general-relativistic weak-field effects within rotating tori: the dynamic anti-dragging and the deviation from the Keplerian motion induced by the fluid selfgravity.

We show that the solutions obtained in the paper 'An exact solution for arbitrarily rotating gaseous polytropes with index unity' by Kong, Zhang, and Schubert represent only approximate solutions of the free-boundary Euler-Poisson system of equations describing uniformly rotating, self-gravitating polytropes with index unity. We discuss the quality of such solutions as approximations to the rigidly rotating equilibrium polytropic configurations.