The conversion of 31 to 34 is accompanied by the formation of an isomer (35). Provide a mechanism for formation of this isomer (Figure 7) and explain (along with a mechanism) (Figure 8) how the authors were able to recycle the undesired isomer.

Figure 7. Mechanism for Formation of 35.

The mechanism for recycling the undesired isomer using DMAP in refluxing toluene is pictured below.  Since the literature indicates that the ratio of the desired isomer 34 to the undesired isomer 35 is exactly the same after this recycling mechanism and after the initial reaction that results in the two isomers, it was determined that the original epoxide must be reformed. 
DMAP functions as both a nucleophile and a proton acceptor/donor in the mechanism.  DMAP attacks the carbonyl, forming a tetrahedral intermediate.  A lone pair-assisted ionization results in the reformation of the carbonyl and a negative charge on the oxygen that was ejected from the tetrahedral intermediate.  The reformed carbonyl undergoes an additional lone pair-assisted ionization in order to reform the DMAP and form an acylium ion.   Simultaneously, the negatively-charged oxygen attached to the six-membered ring attacks the ring at the position where the ring carbon is bonded to a hydroxyl group.  The original epoxide is reformed when the hydroxyl group leaves and attacks the acylium ion, reforming acetic acid.  Once the original epoxide is formed and acetic acid is regenerated, the reaction proceeds through the same steps as the original reaction; the only difference is that DMAP acts as the proton acceptor/donor rather than triethylamine.

Figure 8. Mechanism for Recycling the Undesired Isomer 35⁴.

References:

1. Kim, J. W.; Cho, D. W.; Park, G.; Kim, S. H.; Ra, C. S. Bull. Korean Chem. Soc. 2013, 34, 2286-2290.
2. Li, X.; Li, G.; Chang, H.; Zhang, Y.; Wei, W. RSC Adv. 2014, 4, 6490-6495.
3. Bukowska, A.; Bukowski, W.; Noworól, J. J. Appl. Polym. Sci. 2010, 117, 655-663.