Galilei symmetries of KdV-type nonlinear evolution equations

•All inequivalent KdV-type nonlinear evolution equations admitting the classical Galilei groups are constructed.•All extended Galilei transformation groups and corresponding PDEs are listed.•All natural extensions of the extended Galilei algebras and their invariant equations are described.•Invariant equations obtained may be used as motion equations for they all satisfy the Galilei relativity principle.•A complete classification of group invariant solutions for one obtained Galilei invariant equation is achieved.

We perform Galilei symmetry group classification of a class of third-order nonlinear evolution equations in one spatial variable, which generalize KdV and mKdV equations. All inequivalent PDEs belonging to the class in question which admit the classical Galilei group, the extended Galilei group and the natural extensions of the extended Galilei groups are constructed. The list of so obtained invariant equations may be used as motion equations for they all satisfy the Galilei relativity principle. In addition, we also give a complete classification of group invariant solutions for one obtained Galilei-invariant equation.