  We explore the application of generating symmetries, i.e. symmetries that
depend on a parameter, to integrable hyperbolic third order equations, and in
particular to consistent pairs of such equations as introduced by Adler and
Shabat (AS). Our main result is that different infinite hierarchies of
symmetries for these equations can arise from a single generating symmetry by
expansion about different values of the parameter. We illustrate this, and
study in depth the symmetry structure, for two examples. The first is an
equation related to the potential KdV equation taken from AS. The second is a
more general hyperbolic equation than the kind considered in AS. Both equations
depend on a parameter, and when this parameter vanishes they become part of a
consistent pair. When this happens, the nature of the expansions of the
generating symmetries needed to derive the hierarchies also changes.
