  The $b$-family is a one-parameter family of Hamiltonian partial differential
equations of non-evolutionary type, which arises in shallow water wave theory.
It admits a variety of solutions, including the celebrated peakons, which are
weak solutions in the form of peaked solitons with a discontinuous first
derivative at the peaks, as well as other interesting solutions that have been
obtained in exact form and/or numerically. In each of the special cases $b=2,3$
(the Camassa-Holm and Degasperis-Procesi equations, respectively) the equation
is completely integrable, in the sense that it admits a Lax pair and an
infinite hierarchy of commuting local symmetries, but for other values of the
parameter $b$ it is non-integrable. After a discussion of travelling waves via
the use of a reciprocal transformation, which reduces to a hodograph
transformation at the level of the ordinary differential equation satisfied by
these solutions, we apply the same technique to the scaling similarity
solutions of the $b$-family, and show that when $b=2$ or $3$ this similarity
reduction is related by a hodograph transformation to particular cases of the
Painlev\'e III equation, while for all other choices of $b$ the resulting
ordinary differential equation is not of Painlev\'e type.
