Greencard Renewal (Potential Issue?): Greencard has compound last name, Social Security Only One

AmerCan

Registered Users (C)
Dry stats:

* Wife arrived in 2002,
* Adjustment of Status Approved and Greencard Issued: 2004
* No civil or criminal issues.
* Wife has worked from 8 years,
* taxes all current,
* We have two boys.

Now here's the kicker: When INS issued her green card they did the name like this First Middle Initial Maiden Married. (Example: Jane J Doe Smith). No hyphenation. We didn't think anything of it. We applied for her social security number they put the name as Jane C Smith, we called they said they don't do compound or hyphenated names, state issued ID card was done the same way "There's no hyphen, so they dropped the maiden for the married". Bank accounts, medical insurance, jobs applications, all of them have went off the SS card and not the GC in terms of using only my last name and not the compound one on the GC; the GC was presented in all cases. She's had immigration, employment verification, criminal, ID verification for civil court matters (Bankruptcy filing), and credit checks--all of them with the GC presented as proof of legal residence and legal work status in combo with the SS card as proof of social security numbers. We've been told that since there is no connecting hyphen (example: doe-smith) and is compound (example: doe smith) that it can't be used a compound name, they just drop her maiden portion and use only my last name.

Question: Is this going to present or could present a problem when she renews in 2014? Or is this something Immigration has or could have dealt with in the past? Or should we start trying to get her SS card changed?
 
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