  In this paper we study symmetry reductions of a class of nonlinear third
order partial differential equations $u_t -\epsilon u_{xxt} +2\kappa u_x= u
u_{xxx} +\alpha u u_x +\beta u_x u_{xx}$ where $\epsilon$, $\kappa$, $\alpha$
and $\beta$ are arbitrary constants. Three special cases of equation (1) have
appeared in the literature, up to some rescalings. In each case the equation
has admitted unusual travelling wave solutions: the Fornberg-Whitham equation,
for the parameters $\epsilon=1$, $\alpha=-1$, $\beta=3$ and $\kappa=\tfr12$,
admits a wave of greatest height, as a peaked limiting form of the travelling
wave solution; the Rosenau-Hyman equation, for the parameters $\epsilon=0$,
$\alpha=1$, $\beta=3$ and $\kappa=0$, admits a ``compacton'' solitary wave
solution; and the Fuchssteiner-Fokas-Camassa-Holm equation, for the parameters
$\epsilon=1$, $\alpha=-3$ and $\beta=2$, has a ``peakon'' solitary wave
solution. A catalogue of symmetry reductions for equation (1) is obtained using
the classical Lie method and the nonclassical method due to Bluman and Cole.
