The tame quotient of the inertia group of $\mathbf Q_p$, say, is the profinite group generated by the Frobenius $\sigma$ and the *monodromy* $\tau$, subject to the relation $\tau^{p-1} [\tau, \sigma] = 1$.

Question: why is this second generator called a monodromy, or, more precisely what O.D.E. corresponds to this situation?

Related question: Katz, in "On the calculation of some differential Galois groups" suggests to think of the inertia group as being analogous to the Tannaka group of some category of certain $D$-modules. How does that idea relate to the above situation?

Thank you!