I am looking for examples (references) of pairs of non Quillen-equivalent model categories having the same homotopy categories. 


The motivation is of course that I have two model categories and all the attempts to prove that they are Quillen equivalent have failed so far, but the homotopy categories are equivalent as category. I cannot find in this case any zig-zag of adjunctions between them. Reading such examples could help me to understand better the problem.



Examples of non Quillen-equivalent model categories having equivalent homotopy categories
The ones which usually come up to my mind as soon as I think of this:


* The categories of modules over $\mathbb Z/p^2$ and $\mathbb F\_p[\epsilon]/(\epsilon^2)$, $p$ a prime integer. These rings are quasi-Frobenius, so their module categories have a model structure where cofibrations are monomorphisms, fibrations are epimorphisms and weak equivalences are homomorphisms which become isomorphisms in the stable module category, obtained from the module category by killing projective-injective objects. The homotopy category is this stable module category, which in both cases is the category of $\mathbb F\_p$-vector spaces. This example is equivalent to Rasmus'.
* The category of DG-modules over $\mathbb F\_p[v\_n^{\pm1}]$, $|v\_n|=2p^n-2$, $d(v\_n)=0$, and the category of modules over Morava's $K(n)$.
* The model category of spectra localized at $K\_{(p)}$-equivalences, where $K$ is complex $K$-theory and $p$ is an odd prime, and Franke's algebraic model category $C^{(T,N)}(\mathcal A)$ defined in <http://www.math.uiuc.edu/K-theory/0139/>. This doesn't happen for $p=2$ by results of Constanze Roitzheim.


In all these cases the model categories are stable and the homotopy categories are not only equivalent as categories but as *triangulated* categories.


