Symmetry reductions of the following class of nonlinear third-order partial differential equations
with four arbitrary constants are considered. This class has previously been studied by C. Gilson and A. Pickering [Phys. A, Math. Gen. 28, 2871-2888 (1995; Zbl 0830.35127)] using Painlevé theory. It contains as special cases the Fornberg-Whitham, the Rosenau-Hyman, and the Camassa-Holm equation. The authors apply besides the standard symmetry approach also the non-classical method of G. W. Bluman and J. D. Cole [J. Math. Mech. 18, 1025-1042, (1969; Zbl 0187.03502)]. Using the so-called differential Gröbner bases developed by one of the authors they obtain a symmetry classification of the parameters . The computations are done with the help of the Maple package.