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?
There are some US universities whose requirements are (a) pass the qualifying examinations and (b) write a dissertation acceptable to the faculty. You need to find such a place by searching and talking to people. You need to find a supervisor who is happy with your work and may accept it as is or with some additional work. But you probably also need to pass a set of qualifiers in several math areas. This may be easy for you or not.
But finding the supervisor who thinks this is a good idea is the main step. Best will be someone interested in becoming a collaborator and who has a definite interest in what you have already done.
But there will be more than paperwork involved.
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.
If your completed work is not enough for a PhD dissertation in a US program, it's likely that you would need to take at least some exams and fulfill other requirements like any other student in a math PhD program in the US. But if your work is (1) enough for a PhD and (2) really exceptional, it is possible that some programs would grant the PhD with a minimum of additional requirements. The only way to know is to ask.