A gauge-theoretic description of $\mu$-prolongations, and $\mu$-symmetries of differential equations

We consider generalized (possibly depending on fields as well as on space-time variables) gauge transformations and gauge symmetries in the context of general – that is, possibly non variational nor covariant – differential equations. In this case the relevant principal bundle admits the first jet bundle (of the phase manifold) as an associated bundle, at difference with standard Yang-Mills theories. We also show how in this context the recently introduced operation of $\mu$-prolongation of vector fields (which generalizes the $\text{la}$-prolongation of Muriel and Romero), and hence $\mu$-symmetries of differential equations, arise naturally. This is turn suggests several directions for further development. S0ome detailed examples are also given.