Motivated by these posts, [An NP-complete variant of factoring][1] and [Relationship between symmetry and computational intractability][2], It seems to be worthwhile to investigate the different factors that increase the hardness of problems in NPI. I'm interested in other NP-complete variants of problems of intermediate complexity. 

Is there a recent survey paper of $NP$-complete variants of problems of intermediate complexity (problems between P and NP-complete)?

Cross posted on [TCS StackExchange][3] 

UPDATE: Bounty may be awarded to an answer that gives other interesting NPI problems that have an $NP$-complete variant.

[1]: http://cstheory.stackexchange.com/q/4769/495
[2]: http://cstheory.stackexchange.com/q/871/495
[3]: http://cstheory.stackexchange.com/questions/6761/np-complete-variants-of-npi-problems