  Steady plasma flows have been studied almost exclusively in systems with
continuous symmetry or in open domains. In the absence of continuous symmetry,
the lack of a conserved quantity makes the study of flows intrinsically
challenging. In a toroidal domain, the requirement of double-periodicity for
physical quantities adds to the complications. In particular, the
magnetohydrodynamics (MHD) model of plasma steady-state with the flow in a
non-symmetric toroidal domain allows the development of singularities when the
rotational transform of the magnetic field is rational, much like the
equilibrium MHD model. In this work, we show that steady flows can still be
maintained provided the rotational transform is close to rational and the
magnetic shear is weak. We extend the techniques developed in carrying out
perturbation methods to all orders for static MHD equilibrium by Weitzner
(Physics of Plasmas 21, 022515 (2014)) to MHD equilibrium with flows. We
construct perturbative MHD equilibrium in a doubly-periodic domain with nearly
parallel flows by systematically eliminating magnetic resonances order by
order. We then utilize an additional symmetry of the flow problem, first
discussed by E. Hameiri in (J. Math. Phys. \textbf{22}, 2080 (1981) Sec. III),
to obtain a generalized Grad-Shafranov equation for a class of non-symmetric
three-dimensional MHD equilibrium with flows both parallel and perpendicular to
the magnetic field. For this class of flows, we are able to obtain
non-symmetric generalizations of integrals of motion, such as Bernoulli's
function and angular momentum. Finally, we obtain the generalized Hamada
conditions, which are consistency conditions necessary to suppress singular
currents in such a system when the magnetic field lines are closed. We do not
attempt to address the question of neoclassical damping of flows.
