We propose two novel solvers for estimating the egomotion of a calibrated camera mounted to a moving vehicle from a single affine correspondence via recovering special homographies. For the first class of solvers, the sought plane is expected to be perpendicular to one of the camera axes. For the second class, the plane is orthogonal to the ground with unknown normal, e.g., it is a building facade. Both methods are solved via a linear system with a small coefficient matrix, thus, being extremely efficient. Both the minimal and over-determined cases can be solved by the proposed methods. They are tested on synthetic data and on publicly available real-world datasets. The novel methods are more accurate or comparable to the traditional algorithms and are faster when included in state of the art robust estimators.