Last name differences on tax filings

I have dual citizenship. In one country (where I grew up), my formal last name is a compound: 1st my father's last and 2nd mother's last name. Together they are my last name in that country. In the U.S.A. (where I was born and returned to live in, years ago) my passport and SS card only have my father's last name (no compounds).

 

The problem: I started filing USA taxes using only my first last name, as in my official USA documents. But since 2008, I have been filing taxes with my compound last name (as in my other country's citizenship documents). So, the IRS has tax filings with the 2 versions. I have never had a problem. But:

 

1. Do I need to correct this? If so, where do I start?

 2. Will I run into problems when I qualify for medicare if I do not fix this?

 

Thank you for your help!