  Let $\Sigma$ be a compact surface equipped with an area form. There is an
long standing open question by Katok, which, in particular, asks whether every
entropy-zero Hamiltonian diffeomorphism of a surface lies in the $C^0$-closure
of the set of integrable diffeomorphisms. A natural generalization of this
question is to ask to what extent one family of `simple' Hamiltonian
diffeomorphisms of $\Sigma$ can be approximated by the other. In this paper we
show that the set of autonomous Hamiltonian diffeomorphisms is not Hofer-dense
in the set of integrable Hamiltonians. We construct explicit examples of
integrable diffeomorphisms which cannot be Hofer-approximated by autonomous
ones.
