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This text is devoted to hyperbolic equations admitting differential operatorsthat map any function of one independent variable into a symmetry of thecorresponding equation. We use the term `symmetry driver' for such operatorsand 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 theFr\'echet derivative of an integral is also pre-Hamiltonian (in a new set ofthe variables) if both the symmetry driver and the integral have the smallestorders.

Pre-Hamiltonian operators related to hyperbolic equations of Liouville type