We consider the possible concentration in phase space of a sequence of
eigenfunctions (or, more generally, a quasimode) of an operator whose principal
symbol has completely integrable Hamilton flow. The semiclassical wavefront set
$WF_h$ of such a sequence is invariant under the Hamilton flow. In principle
this may allow concentration of $WF_h$ along positive codimension sub-tori of a
Liouville torus $\mathcal{L}$ if there exist rational relations among the
frequencies of the flow on $\mathcal{L}.$ We show that, subject to
non-degeneracy hypotheses, this concentration may not in fact occur. The main
tools are the spreading of Lagrangian regularity on $\mathcal{L}$ previously
shown by Vasy and the author, and an analysis of higher order transport
equations satisfied by the principal symbol of a Lagrangian quasimode.