Analytic series solutions are constructed for the phreatic free surface problem of two-dimensional steady downslope saturated-unsaturated flow, with water exiting at a seepage face. The region in free parameter space is delineated for which the water table intersects the upper surface, and the steady state with uniform constant irrigation rate, ceases to exist. The flow solution is extended to a case of the domain being more general than a parallelogram, with the upper and lower boundaries being piecewise linear. This geometry resembles that of large-scale rainfall simulators that are designed to test slope stability of wetted soil beds.