Finishing a PhD that I started abroad, in the USA? I studied in Europe some times ago but didn't finish my PhD dissertation. Recently I finished it and I published in a math journal. I never studied in the USA, but I have been living and working there. Is it possible to finish this PhD with the same results that I have found, or I have to start all over again?
I disagree with the other answers. Virtually all reputable US graduate schools have a residency requirement. Further, they will have little interest in a "pass-through" candidate who feels that they have already earned a PhD elsewhere and wants to obtain it as quickly as possible. 

Your best bet is to follow up with your old program / contacts. Even retired professors might be willing to work with you. Perhaps someone knows someone who will take a special interest in you. With a professor on your side, rules can be bended or special arrangements made.

If that is a strict no-go, then you would essentially have to start over at a new institution. You could shop for an institution that will minimize the additional work you will need to do, but I would still expect it to take multiple years.