A nonlinear solution to the Grad–Shafranov equation (Cicogna et al 2010 Phys. Plasmas17 102506) is extended to plasmas with incompressible flows parallel to the magnetic field. The solution can describe either crescent-shaped or D-shaped equilibria having hollow current density profiles but safety factors monotonically increasing from the magnetic axis to the plasma boundary. Application of a sufficient condition for linear stability implies that this condition is satisfied in a plasma area located on the high magnetic field side. Stabilization is related to the variation of the magnetic field perpendicular to the magnetic surfaces. The area in which the condition is satisfied depends on the plasma shape but is insensitive to the amplitude and shear of the flow.