  Metric perturbations typically strengthen field resonances during preheating.
In contrast we present a model in which the super-Hubble field resonances are
completely {\em suppressed} when metric perturbations are included. The model
is the nonminimal Fakir-Unruh scenario which is exactly solvable in the
long-wavelength limit when metric perturbations are included, but exhibits
exponential growth of super-Hubble modes in their absence. This gravitationally
enhanced integrability is exceptional, both for its rarity and for the power
with which it illustrates the importance of including metric perturbations in
consistent studies of preheating. We conjecture a no-go result - there exists
no {\em single-field} model with growth of cosmologically-relevant metric
perturbations during preheating.
