Equilibrium eigenstates of an axisymmetric magnetically confined plasma with toroidal flow are investigated by means of exact solutions of the ideal magnetohydrodynamic equations. The study includes "compressible" flows with constant temperature, but varying density on magnetic surfaces and incompressible ones with constant density, but varying temperature thereon. In both cases eigenfunctions of the form Ψ_nl = Z_l(z)R_n(R) (l, n=1,2,...) describe configurations with lxn magnetic axes. By varying the flow parameters a change in magnetic topology is possible. In addition, the effects of the flow and the aspect ratio on the Shafranov shift are evaluated along with the variations of density and temperature on magnetic surfaces.