The paper gives the exact solutions of the equations involved in the axial momentum theory for several kinds of radially variable load distributions. Specifically, a relation between the power and thrust coefficients is obtained for a load distribution of polynomial type well representing physically relevant loads. The paper also analyses simpler loads of the trapezoidal, triangular and linear type. The mass flow swallowed by the rotor is also evaluated for all cases. By providing a set of exact solutions of the axial momentum theory for rather complex load shapes, the study offers the unique opportunity to verify the correctness of existing or new numerical methods based on the actuator disk approach for other than the trivial, though widely used, uniform load solution. (C) 2017 Elsevier Ltd. All rights reserved.