  We prove that the only global strong solution of the periodic rod equation
vanishing in at least one point $(t_0,x_0)$ is the identically zero solution.
Such conclusion holds provided the physical parameter $\gamma$ of the model
(related to the finger deformation tensor) is outside some neighborhood of the
origin and applies in particular for the Camassa--Holm equation, corresponding
to $\gamma=1$. We also establish the analogue of this unique continuation
result in the case of non-periodic solutions defined on the whole real line
with vanishing boundary conditions at infinity. Our analysis relies on the
application of new local-in-space blowup criteria and involves the computation
of several best constants in convolution estimates and weighted Poincar\'e
inequalities.
