  The exhaustive classification of stationary incompressible flows with
constant total pressure of ideal infinitely electrically conducting fluid is
given. By introduction of curvilinear coordinates based on streamlines and
magnetic lines of the flow the system of magnetohydrodynamics (MHD) equations
is reduced to a nonlinear vector wave equation extended by the
incompressibility condition in a form of a generalized Cauchy integral. For
flows with constant total pressure the wave equation is explicitly integrated,
whereas the incompressibility condition is reduced to a scalar equation for
functions, depending on different sets of variables. The central difficulty of
the investigation is the separation of variables in the scalar equation, and
integration of the resulting overdetermined systems of nonlinear partially
differential equations. The canonical representatives of all possible types of
solutions together with equivalence transformations, that extend the canonical
set to the whole amount of solutions are represented.
