Cylindrical ideal magnetohydrodynamic equilibria with incompressible flows

It is proved that (a) the solutions of the ideal magnetohydrodynamic equation, which describe the equlibrium states of a cylindrical plasma with purely poloidal flow and arbitrary cross sectional shape [G. N. Throumoulopoulos and G. Pantis, Plasma Phys. and Contr. Fusion 38, 1817 (1996)] are also valid for incompressible equlibrium flows with the axial velocity component being a free surface quantity and (b) for the case of isothermal incompressible equilibria the magnetic surfaces have necessarily circular cross section.