  The exponential Teichm\"uller spaces $E_p$, $0\leq p \leq \infty$,
interpolate between the classical Teichm\"uller space ($p=\infty$) and the
space of harmonic diffeomorphisms $(p=0)$. In this article we prove the
existence of non-variational critical points for the associated functional:
mappings $f$ of the disk whose distortion is $p$-exponentially integrable,
$0<p<\infty$, yet for {\em any} diffeomorphism $g(z)$ of $\ID$ with
$g|\partial\ID=identity$ and $g\neq identity$ we have $f\circ g$ is not of
$p$-exponentially integrable distortion.
