The author extends the method proposed earlier by himself and W. O. Criminale [Proc. R. Soc. Lond., Ser. A 406, 13-26 (1986; Zbl 0602.76032)] originally to find exact solutions to the Navier-Stokes equations for an incompressible viscous fluid, now to solve the basic equations of magnetohydrodynamics. The incompressible ideal MHD flow is studied in the form given by Elsasser, the new quantities are decomposed in those pertaining to a basic flow and those describing disturbances. The basic solutions are unbounded in space, the disturbances have a plane wave form. The conditions derived from the basic equations both give restrictions to the admissible basic flows and determine amplitudes and wave numbers of the disturbances. The admissible basic flows are discussed which contain cases for constant and spatially varying magnetic fields as well as time-dependent ones. The disturbances which may be superposed upon the admissible flows, are described by linear equations.