Free-surface flow past a disturbance at critical Froude number is commonly found to be unsteady with complex wave patterns both upstream and downstream of the disturbance. Such flows can be undesirable as the waves that are generated can have a negative impact in applications including the erosion of waterway banks and energy loss through wave drag on a ship. This motivates us to develop a new approach to obtain steady solutions at critical Froude number that are wave free in the far field. Under the assumption of two-dimensional, irrotational, incompressible fluid flow, we show that both weakly and fully nonlinear solutions to the problem are non-unique. A range of qualitatively different types of numerical solutions and analytical approximations are discovered, for example for flow over a corrugated channel bottom.