  In this paper we study symmetry reductions and exact solutions of the shallow
water wave (SWW) equation $$u_{xxxt} + \alpha u_x u_{xt} + \beta u_t u_{xx} -
u_{xt} - u_{xx} = 0,\eqno(1)$$ where $\alpha$ and $\beta$ are arbitrary,
nonzero, constants, which is derivable using the so-called Boussinesq
approximation. Two special cases of this equation, or the equivalent nonlocal
equation obtained by setting $u_x=U$, have been discussed in the literature.
The case $\alpha=2\beta$ was discussed by Ablowitz, Kaup, Newell and Segur
[{\it Stud.\ Appl.\ Math.}, {\bf53} (1974) 249], who showed that this case was
solvable by inverse scattering through a second order linear problem. This case
and the case $\alpha=\beta$ were studied by Hirota and Satsuma [{\it J.\ Phys.\
Soc.\ Japan}, {\bf40} (1976) 611] using Hirota's bi-linear technique. Further
the case $\alpha=\beta$ is solvable by inverse scattering through a third order
linear problem. In this paper a catalogue of symmetry reductions is obtained
using the classical Lie method and the nonclassical method due to Bluman and
Cole [{\it J.\ Math.\ Mech.\/}, {\bf 18} (1969) 1025]. The classical Lie method
yields symmetry reductions of (1) expressible in terms of the first, third and
fifth \p\ transcendents and Weierstrass elliptic functions. The nonclassical
method yields a plethora of exact solutions of (1) with $\alpha=\beta$ which
possess a rich variety of qualitative behaviours. These solutions all like a
two-soliton solution for $t<0$ but differ radically for $t>0$ and may be viewed
as a nonlinear superposition of two solitons, one travelling to the left with
arbitrary speed and the other to the right with equal and opposite speed.
