I have done alot of research into what can be towed by my 2009 Santa Fe Limited, which has the 3.3 liter and tow package. There is alot of infomation out there with regards to using a Weight Distribution Hitch (WDH). I have spoke with 2 Hyundai service managers at different locations, Curt (hitch Mfg), and RV service manager with several years experience.
The results are as follows to allow a WDH:

• if your tongue wt. is below the maximum allowed wt of 350lbs.
• Total trailer of 3500lbs.

The idea of using a WDH is to bump up the amount of tongue weight and trailer weight, as it transfers tongue weight onto the full tow vehicle (TV).
I was wanting to make use of the anti-sway bay within the TV's capacity, and is now been confirmed the Santa Fe is capable of using this set-up.