We propose 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, second and third classes of solvers, the sought plane is expected to be perpendicular to one of the camera axes. For the fourth class, the plane is orthogonal to the ground with unknown normal, e.g., it is a building facade. All 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 solvers. 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. The source code is publicly available.