  This text is devoted to hyperbolic equations admitting differential operators
that map any function of one independent variable into a symmetry of the
corresponding equation. We use the term `symmetry driver' for such operators
and prove that any symmetry driver of the smallest order is pre-Hamiltonian
(i.e., the image of the driver is closed with respect to the standard bracket).
This allows us to prove that the composition of a symmetry driver with the
Fr\'echet derivative of an integral is also pre-Hamiltonian (in a new set of
the variables) if both the symmetry driver and the integral have the smallest
orders.
