Are there any non-Jahn-Teller type distortions? My question is essentially, if ANY gradual type of deviation from a higher possible symmetric arrangement of the nuclei in a molecule is covered by the Jahn-Teller model. 

More specifically I have in mind some lets say "ionic complexes" between a very small cation and some larger (possibly weakly coordinating) anions. 

Very specifically, I remember I have seen for example a calculated equilibrium structure of a beryllium bis-2-fluorobenzoate $\ce{(Be[o-C6H4FCOO]2}$, isomer with $\ce{F\bond{...}Be-O}$ coordination), which showed a distinct deviation from $\ce{C2}$ symmetry. The only type of JT-distortion this would qualify for is 2nd order one. But then since its largely an ionic species I suppose the HOMO-LUMO gap should be so large that 2nd order JT is very unlikely (I am aware that $\ce{C2}$ is not Jahn-Teller active but through vibronic coupling of close-lying $\ce{^2A}$ and $\ce{^2B}$ also JT-like distortions can occure). 

I imagine that if the cation is very small so that it "dangles" in the gap between the anions it might be energetically more favourable if the cation decides to stick on one than to balance in the centre. Just a naive picture. But would such a thing - if it exists - be subsumed under JT as well?

So, is distortion possible for ionic aggregates and would such a thing also be described by JT?
Apparently there is some theorem e.g. to be found in an article from Isaak B Bersuker with title "SPONTANEOUS SYMMETRY BREAKING IN MATTER INDUCED BY DEGENERACIES AND PSEUDODEGENERACIES" in Advances in Chemical Physics, Volume 160, First Edition. Stuart A. Rice and Aaron R. Dinner. © 2016 John Wiley &amp; Sons, Inc. Published 2016 by John Wiley &amp; Sons, Inc; p 187-

The theorem reads like: 

"DEGENERACY AND PSEUDODEGENERACY ARE THE ONLY SOURCE OF SPONTANEOUS SYMMETRY BREAKING (SSB) IN ATOMIC SYSTEMS"

And can be reformulated by:

" [...] can we get structural instability of high‐symmetry configurations in any electronic state without involving interaction with other states [...]? The answer to these questions is no: any instability and SSB in polyatomic systems leading to their formation, deformation, and transformation involves at least two electronic states, degenerate or pseudodegenerate. This is the two‐state paradigm in structural instability."

So in essence that means all types of molecular distortions can be traced back to some JT effect.