  We perform a complete study by using the theory of invariant point
transformations and the singularity analysis for the generalized Camassa-Holm
equation and the generalized Benjamin-Bono-Mahoney equation. From the Lie
theory we find that the two equations are invariant under the same
three-dimensional Lie algebra which is the same Lie algebra admitted by the
Camassa-Holm equation. We determine the one-dimensional optimal system for the
admitted Lie symmetries and we perform a complete classification of the
similarity solutions for the two equations of our study. The reduced equations
are studied by using the point symmetries or the singularity analysis. Finally,
the singularity analysis is directly applied on the partial differential
equations from where we infer that the generalized equations of our study pass
the singularity test and are integrable.
