  Many magnetic structures in the solar atmosphere evolve rather slowly so that
they can be assumed as (quasi-)static or (quasi-)stationary and represented via
magneto-hydrostatic (MHS) or stationary magneto-hydrodynamic (MHD) equilibria,
respectively. While exact 3D solutions would be desired, they are extremely
difficult to find in stationary MHD. We construct solutions with magnetic and
flow vector fields that have three components depending on all three
coordinates. We show that the non-canonical transformation method produces
quasi-3D solutions of stationary MHD by mapping 2D or 2.5D MHS equilibria to
corresponding stationary MHD states, i.e., states that display the same field
line structure as the original MHS equilibria. These stationary MHD states
exist on magnetic flux surfaces of the original 2D MHS states. Although the
flux surfaces and therefore also the equilibria have a 2D character, these
stationary MHD states depend on all three coordinates and display highly
complex currents. The existence of geometrically complex 3D currents within
symmetric field-line structures provide the base for efficient dissipation of
the magnetic energy in the solar corona by Ohmic heating. We also discuss the
possibility of maintaining an important subset of non-linear MHS states, namely
force-free fields, by stationary flows. We find that force-free fields with
non-linear flows only arise under severe restrictions of the field-line
geometry and of the magnetic flux density distribution.
