It is shown that the ideal magnetohydrodynamic (MHD) equilibrium states of an axisymmetric plasma with incompressible flows are governed by an elliptic partial differential equation for the poloidal magnetic flux function ψ containing five surface quantities along with a relation for the pressure. Exact equilibria are constructed including those with nonvanishing poloidal and toroidal flows and differentially varying radial electric fields. Unlike the case in cylindrical incompressible equilibria with isothermal magnetic surfaces which should have necessarily circular cross sections [G. N. Throumoulopoulos and H. Tasso, Phys. Plasmas 4, 1492 (1997)], no restriction appears on the shapes of the magnetic surfaces in the corresponding axisymmetric equilibria. The latter equilibria satisfy a set of six ordinary differential equations which for flows parallel to the magnetic field B can be solved semianalytically. In addition, it is proved the nonexistence of incompressible axisymmetric equilibria with (a) purely poloidal flows and (b) nonparallel flows with isothermal magnetic surfaces and |B|=|B|(ψ) (omnigenous equilibria).

The following theorem is proved in the article: An MHD-unstable configuration with a dissipationless plasma surrounded by vacuum and possibly superconducting walls cannot be stabilized by introducing walls of finite electrical conductivity.

A sufficient condition for the linear stability of three dimensional equilibria with incompressible flows parallel to the magnetic field is derived. The condition involves physically interpretable terms related to the magnetic shear and the flow shear.

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, 1817Fusion 38, (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.

The cat-eyes steady state solution in the framework of hydrodynamics describing an infinite row of identical vortices is extended to the magnetohydrodynamic (MHD) equilibrium equation with incompressible flow of arbitrary direction. The extended solution covers a variety of equilibria including one-and two-dimensional generalized force-free and Harris-sheet configurations which are preferable from those usually employed as initial states in reconnection studies. Although the vortex shape is not affected by the magnetic field, the flow in conjunction with the equilibrium nonlinearity has a strong impact on isobaric surfaces by forming pressure islands located within the cat-eyes vortices. More importantly, a magnetic-field-aligned flow of experimental fusion relevance and the flow shear have significant stabilizing effects in the region of pressure islands. The stable region is enhanced by an external axial ("toroidal") magnetic field.1