*(English)*Zbl 0879.35005

Symmetry reductions of the following class of nonlinear third-order partial differential equations

with four arbitrary constants $\epsilon ,\kappa ,\alpha ,\beta $ 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 $\epsilon ,\kappa ,\alpha ,\beta $. The computations are done with the help of the Maple package.

##### MSC:

35A25 | Other special methods (PDE) |

58J70 | Invariance and symmetry properties |

13P10 | Gröbner bases; other bases for ideals and modules |

35Q58 | Other completely integrable PDE (MSC2000) |

37J35 | Completely integrable systems, topological structure of phase space, integration methods |

37K10 | Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies |

68W30 | Symbolic computation and algebraic computation |